Are relative simultaneity and absolute simultaneity equally consistent with empirical observation? I have long been skeptical, from a strictly conceptual standpoint, of the notion of relative simultaneity.  I am reading a recent (2021) paper which, using the sagnac effect as a basis, claims that relative simultaneity is inconsistent with experiment.  Here is the link to that paper:  https://www.tandfonline.com/doi/full/10.1080/09500340.2021.1887384
This paper contains a number of references and has been cited twice.  In some respects it is quite technical, being chock-full of mathematical formulae and calculation.  I will now post a few excerpt from the paper which I found interesting.   These excerpts pertain primarily to the methodology and the conclusions of the paper, not the specific basis for the conclusions.  First, from the abstract:

We thoroughly examine the role of absolute and relative simultaneity in the interpretation of the Sagnac effect  using an approach that allows for determining the local speed along the light path...a rigorous and coherent interpretation of the Sagnac effect favours absolute over relative simultaneity...The implications for the Lorentz transformations and synchronization by means of the Global Positioning System are considered.

From the main body of the paper:

Authors adhering to the conventionalist thesis [2,4,10,12–15] assume
that the LT and LTA are physically equivalent. However, the
equivalence of Einstein and absolute synchronization implies the
equivalence of relative and absolute simultaneity, while the two are
conceptually incompatible. ...
[We are] using an approach that assumes the non-equivalence of
absolute and relative simultaneity and takes into account the decisive
role of clock synchronization.
Moreover, a recent work has shown that relative and absolute
simultaneity (and the LT and LTA) can be discriminated experimentally
with the implication that the one-way speed of light is measurable in
principle and, therefore, the conventionality of the speed of light no
longer holds.
For determining the value of the local light speed, in this paper we
use a novel approach that emphasizes the essential roles of absolute
and relative simultaneity and the related clock synchronization
procedures in the interpretation of the Sagnac effect.
Then, taking rigorously into account clock synchronization, our
approach reveals the inconsistencies that emerge by requiring the
local light speed to be c in both the linear and circular Sagnac
effect, here considered in the context of Relativity Theory.
We highlight that the inconsistencies we found with relative
simultaneity can be related with the ‘undesirable consequences’
pointed out recently by Lee [23] for light propagation in (closed)
cylindrical spacetime. As already shown by Selleri [3], by us [6], and
now by Lee, these inconsistencies, or undesirable consequences,
disappear when absolute synchronization is adopted.
The many attempts aimed to rebut Sagnac's rational claim are quite
understandable from some perspectives, although it is unreasonable to
try to justify and preserve the paradigm of the constancy of c at any
cost, for example, by claiming that synchronization is conventional.
Considering that Sagnac's effect can be coherently interpreted with
absolute simultaneity, then it is difficult to support the
conventional arguments that Einstein synchronization is also confirmed
[12–15,27] when relative simultaneity is met with the discontinuity of
the time gap.
...Einstein synchronization fails when applied to a closed contour
[6,8,9,16–18,20] and this unphysical prediction of a discontinuity in
time on a rotating disc arising from Einstein synchronization has been
appropriately described by Klauber [17] as ‘bizarre’....Einstein
synchronization, or more precisely the equivalent assumption of the
invariance of c, leads to a result in disagreement with observation.
...since the Sagnac effect can be described coherently by means of
absolute simultaneity, but not with relative simultaneity, absolute
and Einstein synchronization cannot be physically equivalent, as has
been proved independently in the work of [5]...
...for several decades there has been a recognition, visible in recent
literature [6,8,9,12,16–18,23] that conservation of simultaneity and
the related transformations offer a simpler and physically meaningful
way to interpret optical experiments.
The conclusion is that a coherent interpretation of the Sagnac effect
favours absolute over Einstein synchronization, indicating that
transformations based on absolute simultaneity (such as the LTA) are
likely candidates for describing the whole body of natural phenomena."

As I said, I have mainly just quoted conclusory statements.  I have little doubt that advocates of relative simultaneity would argue with these conclusions until the cows come home.
That said, does anyone here have comments on the merits of this paper?  Are there mathematical mistakes, interpretative errors, insufficient "evidence," illogic, for other flaws in this paper which would negate its conclusions?  I have not read all the papers referred to in this paper, which would presumably allow for a more complete understanding, but maybe one of you has (or will).
 A: I have read the first page of the article you mention in your answer to your own question. I did not read any further because the first page alone contained sufficient evidence that its author's logic is faulty. You should bear in mind that the internet is littered with so called refutations of special relativity. I have examined a few of them in detail, each of which revealed that their author imply mis-understood how SR works. The paper you mentioned contains a number of obvious mistakes. The author asserts, for example, that the physical properties of a moving object change owing to the acquisition of additional KE, which in itself is an absurdity, since the moving object could be one that has slowed down relative to the speed of its apparently stationary surroundings and thus lost KE. To critique the entire paper in detail would take far too much effort.
A: @MarcoOcram has answered the question well, but I'll give you my own perspective:
(1) As a general comment: relativity is a well studied and firmly established theory, more than a century old. It's not going to be disproved by thought experiment or by uncovering a logical flaw. If that were the case, that would have happened by now, and almost certainly would have happened in the early days. After all, in 1904 nobody had even heard of relative simultaneity, and absolute simultaneity was the default position. Einstein's work (supplemented by the work of many, many other physicists and mathematicians) had to overcome that default, and it did so.
(2) As for the paper you referenced: it's a "solution" in search of a problem. The whole issue they raised (the different speed of light in different directions in a rotating reference frame) is a strawman, because the assumption in relativity is that the speed of light is constant in an inertial reference frame, which a rotating reference frame definitely is not (as even the authors of the paper acknowledge). They try to dodge that by introducing a "linearized" version of the Sagnac effect, but the reference frame they use for that isn't inertial either. The Sagnac effect is well studied and understood in relativity. The paper really offers nothing of value that I can see, and I'm surprised it was published. But the process of peer review is imperfect, so things like this happen sometimes.
(3) To answer the specific question in the title of your post: yes, both absolute and relative simultaneity can be consistent with experiment. (For the absolute version google "Lorentz Ether Theory".) The trouble is that the "absolute" time is undetectable, and hence so is "absolute" simultaneity. Real clocks moving relative to one another tick at different rates, and it's impossible to determine which shows the "true" time. So physicists have generally discarded the notion of absolute time as superfluous.
A: I disagree with the suggestion that the Sagnac effect presents a challenge to Special Relativity. However, it is indeed the case that the Sagnac effect offers a unique window onto the foundations of theory of motion.

Metric of spacetime
The fundamental assumption that underlies special relativity is that the nature of spacetime is such that it is described by the Minkowski metric.
More generally, the fundamental realization that came with the introduction of special relativity is that for any theory of motion, in order to formulate the theory at all, it is necessary to have a metric of the spacetime in place.
Retroactively we recognize that newtonian mechanics is based on assuming a metric, we can refer to that metric as Galilean metric, in analogy with the concept of Galilean transformation. In newtonian mechanics this was an implicit assumption.
In terms of General Relativity: the metric of spacetime is arrived at by solving the Einstein Field Equations for the specific conditions of the case at hand.

Sagnac effect
The Sagnac effect has been confirmed multiple times, we can regard the Sagnac effect as a given.
As I mentioned in a comment to this question: the best discussion, by far (that I am aware of) is the one by Olaf Wucknitz:
Sagnac effect, twin paradox and space-time topology - Time and length in rotating systems and closed Minkowski space-times
The Sagnac effect does not present a logical challenge to the Minkowski metric as axiom of Special Relativity.
(There are presentations of Special Relativity  in circulation that are over-extended. Over-extended in the sense that more is asserted than is logically necessary. Such an over-extended presentation is seen to be refuted by the existence of the Sagnac effect. The issue is resolved by switching to the appropriate way of presenting special relativity.)
Importantly, the outcome of a Sagnac experiment does not distinguish between galilean metric and Minkowski metric. That is, both in terms of galilean metric and in terms of Minkowski metric the predicted outcome is the same. The Sagnac effect is quite unique in that regard.

Synchronized world time
All over the world there are centers for metrology, that include departments dedicated to accurate time-keeping.
These centers exchange timing signals to maintain a Earth-encompassing synchronized time. The accuracy of that synchronization is in the order of nano-seconds. That level of accuracy is necessary for scientific purposes.
The engineers, working world-wide, have come up with protocols to maintain that world sychronized time. These engineers are, of course, aware of the ramifications of relativistic physics.
These centers of time-keeping are not regularly spaced over the globe of course. For the purpose of simplification the following animation represents a scenario of 4 time keeping centers, spaced evenly along the Equator.

The grey dots represent four time-keeping stations, co-rotating with a rotating platform.
The blue dots and red dots represent timing signals, propagating at the speed of light. Each colored dot can be thought of as a propagating pulse.
The animation represents th following  protocol: the four time-keeping stations are relaying the timing pulses in such a way that they maintain an even spacing of the pulses.
In the animation the blue dots and red dots move at the same rate; the crossing points do not drift, the crossing points remain at the same orientation with respect to the loop.
Notice that for the blue dots the time it takes to travel from one station to the next is longer than for the red dots, because the stations are in circumnavigating motion.
Therefore in order to arrive at a global synchronized time the protocols must take that difference into account. In other words: the protocols for maintenance of global synchronized time have to take the Sagnac effect into account. Otherwise a global synchronized time cannot be maintained.

So we have that the two counterpropagating loops of pulses are acting as a reference of simultaneity.
In both directions the timing pulses are propagating at the speed of light. As a matter of principle the speed of light is the same in all directions.
