The best pessimistic response I read to this experiment when it first came out stated that overturning well-established theorems in physics should be left to experiments where the result is clearly and simply binary, not where the result is an extraordinarily precise measurement in a complex system.
Otoh, one of the early tests for general relativity was a minor deviation in Mercury's orbit from what was predicted by classical mechanics. My rough intuition is that inconsistencies can show up initially as faint flickers, but once you investigate more closely, one can find interesting and prominent counter examples.
I'm not an expert on this, but I believe the deviation in Mercury's orbit was easily observed and calculated, and thus became in a sense a binary call-out of Newtonian physics. Until highly precise measurements of relativistic velocities is common place, experiments like this are of course valuable and interesting, but one instance of an experimental result like this cannot be relied upon to supersede special relativity.
Anything which is going faster than C in one inertial frame of reference, is going backwards in time in another inertial frame of reference, which allows for a violation of causality. I'm sure the experimental set-up would be complicated as balls, but a qualitative property like causality is probably more robust to measurement.
One small disagreement with you. While general relativity suggests that going faster than C causes a violation of causality, it might be the case that things can go faster than C without causing violation of causality, and we need a new model for such cases.
Part of the point of experimentation is to find holes in our existing models, so it is reasonable for scientists to look for faster than C without violation of causality.
It does not necessarily allow for a violation of causality.
As an example, suppose that we were able to go faster than C in the reference frame of the fixed distant stars, but not in other reference frames. There are reference frames where you can go backwards in time, but none in which you can violate causality.
(That said, General Relativity allows for causality violations. However setting them up is well beyond any engineering ability our species is likely to develop...)
