I really doubt it. People are fairly convinced that faster-than-light communication can't happen -- and if it can, it's going to take a bigger change than this.
afaik, causality is still a strong axiom, so no.
The "causality axiom" is a nice (or trivial) statement in philosophy. But as far as we know, there's no such axiom in any mathematical theory proved to match physical reality, and nothing so clear cut in modern theoretical physics at all afaik...
Also, physics isn't that much built on axioms, in general: https://www.youtube.com/watch?v=YaUlqXRPMmY (Summarizing Feynman: you still get from time to time cases where you can prove something from "axioms", but then you also find it to still hold in contexts where the axioms don't apply, or even to not work in others where they do apply... and the latter is the case where you have to remake your theory to match reality)
But my main point is that considering something impossible simply because our puny ape brains can't fathom them is preposterous!
(We'd still have reproducible experiments and so forth in every context that didn't involve whatever extremely weird circumstances would produce causality violations -- and the circumstances would certainly have to be extremely weird.)