I have a migraine and, like @knzhou, don't want to read the paper in detail. Maybe I'll come back to it. But I want to say something about how quantum mechanics works.
In the mathematics of quantum mechanics, you have wavefunctions or quantum states, and you have "observables". In the original, sensible, self-consistent interpretation of quantum mechanics, the observables are what is real, and the wavefunctions and quantum states are a way to predict the values of observables.
The confusion entered when people started talking about wavefunctions as what is real. This was possible, first of all, because there are wavefunctions (called eigenstates) which imply that a specific observable (position, momentum, whatever) takes a specific value with 100% probability. So it became easy to say that "the quantum state is in the position eigenstate for x=x0" is the same thing as "the particle is at the position x=x0".
But then once you say the quantum state is a physical thing, then you will also be ascribing reality to something like "the superposition of the position eigenstate for x=x0, with sqrt(-1) times the position eigenstate for x=x1".
So suppose we don't go there. We stick to the original mathematical division of labor: observables are the only thing that's real, wavefunctions are just a tool of calculation. The next question might be, which observables actually take values, and when and where do they do so?
This is a question because not all possible observables can be real at the same time, as a consequence of the uncertainty principle. The Copenhagen interpretation set a minimum level of reality: at least those things that are measured, must exist in some sense. If they weren't real, they couldn't have been measured.
However, you don't actually have to define quantum mechanics in this observer-centric way. Something called "consistent histories" gives you a mathematical method for assigning quantum-mechanical probabilities to very general sequences of observables. You can posit that there is plenty of unobserved reality, just so long as the actually real observables are "decohered" with respect to each other.
For some reason this ontological option is relatively unexplored. But it definitely nullifies the argument for many worlds in this paper. I didn't get into the details, but clearly their argument involves saying that observers are made of wavefunctions or inhabit wavefunctions, and can engage in coherent interactions.
If you insist that observables are the only real things, and that they must be decohered, then you simply can't have an "observer in a quantum coherent state". If you did try to engineer such a thing, then according to an observables-only ontology, there would just be a gap in the universe's actual history for the duration of the coherence, rather than the many-worlds scenario of multiple copies of an observer somehow coexisting.
The closest they come to addressing this possibility is when they mention "collapse theories", which they classify as a modification of quantum mechanics. A collapse theory indeed resembles a "consistent history" in which wavefunctions and observables are both physical, and the observables are held to result from wavefunction collapse into an eigenstate. (This formal affinity between consistent histories and collapse theories is another fact that seems to have largely escaped notice.)