Will the ever accelerating space expansion (like at the level of inflation) eventually break causality? I have read this question:

requires that "for an action at one point to have an influence at another point, something in the space between the points, such as a field, must mediate the action". In view of the theory of relativity, the speed at which such an action, interaction, or influence can be transmitted between distant points in space cannot exceed the speed of light.

How to understand locality and non-locality in Quantum Mechanics?
As far as I understand, space is expanding at an ever accelerating rate, and it could eventually reach the level of the former inflation right after the big bang. Inflation, and its space expansion was so extreme that it was able to separate otherwise bound/linked systems like particle antiparticle pairs.
At this level of expansion, the speed at which space is expanding can exceed the speed of light (speed of causality), and this might even be true for the space inbetween bound quantum objects, like quarks in a nucleon or electrons and protons in an atom.

From the point of view of one such object, the spacetime is something like an inside-out Schwarzschild black hole—each object is surrounded by a spherical event horizon. Once the other object has fallen through this horizon it can never return, and even light signals it sends will never reach the first object (at least so long as the space continues to expand exponentially).

https://en.wikipedia.org/wiki/Inflation_(cosmology)
So quantum objects are separated by an event horizon, and they keep separating faster then the speed of light (causality), thus the fields inbetween them cannot transmit causality.
The answer says that the fields have to mediate causality inbetween these quantum objects, but if space itself is expanding faster then light, then the field itself will not be able to catch up to space expansion (for example the speed at which otherwise bound elementary particles are flying apart) so causality will not be transmitted.
So basically what I am asking is, does space expansion expand (stretch) the fields that propagate causality?
Question:

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*Will the ever accelerating space expansion (like at the level of inflation) eventually break causality?

 A: In General Relativity, you should only ever compare the speeds of two objects that are (at least approximately) in the same locally inertial frame. In other words, you only compare the speed of two objects if the distance between the objects is much smaller than the length scale over which the spacetime curvature is changing.
Or -- in the context of inflation -- it makes no sense to compare the relative speed of two observers who are separated by a Hubble distance (or more).
Locally near each observer, you can define a light cone (if you treat gravity semi-classically), and quantum fields on that background spacetime will commute if they are spacelike separated, just as in flat-space quantum field theory.
If there is a spacelike hypersurface (eg if we take a "snapshot" of the Universe at a fixed time), then two quantum field operators at different points on this surface will commute.
Additionally, the horizon means that if comoving points $A$ and $B$ are separated by more than one horizon distance at time $t$, then the future light cones of $A$ and $B$ will never intersect. So quantum fields will never be in causal contact, once they are outside each other's horizons.
The $\delta N$ formalism (see, eg: https://arxiv.org/abs/1003.5057, https://arxiv.org/abs/astro-ph/0506262, https://arxiv.org/abs/astro-ph/9507001) in inflation theory takes advantage of this behavior to calculate the statistics of primordial fluctuations in patches of the inflationary Universe that are causally separated from each other.
On a different note, these issues around causally disconnected regions in an inflating Universe creates several problems with formulating quantum field theory on de Sitter spacetime (a fancier name for an exponential inflating Universe). We normally like to talk about the "S-matrix" in flat-space quantum field theory, where $N$ particles come in from infinity, scatter, and go out as $M$ outgoing particles approaching future infinity. The problem is that these outgoing particles will all eventually become separated from each other into different "Hubble patches", each one outside the horizon of the others, so there is no observer at asymptotic future infinity who can "collect" all the outgoing particles and observe what the final state of the scattering experiment was. Therefore it's not clear how to define an S-matrix in de Sitter. Even though you didn't ask about this specifically, I think your question is getting at a deep problem reconciling quantum field theory an de Sitter space; the S-matrix is one clear manifestation at that.
