Fitzgerald and Lorentz showed how if we assume that the structure and dynamics of all matter arises from electromagnetic forces which obey Maxwell’s equations in some particular (“aether”) frame of reference (not necessarily that of the lab itself), then the result would be that moving bodies experience length contraction, slowed vibration, and increased inertial mass – all in such a way that a moving observer would be unable to detect any of these effects on itself and would instead think that objects stationary with respect to the aether were exhibiting them instead.

All experimental results so far (and also, I am sure, the modified Hafele-Keating that I suggested) are consistent with the Fitzgerald-Lorentz prediction of undetectability of the aether and symmetric apparent effects of length contraction, time dilation, and increased inertial mass.

I thought your question was about the symmetry of the situation rather than the existence of a special “aether” frame.

But if you are asking whether any experiment can prove the absence of an aether frame the answer is no. The reason we reject the assumption of an aether frame is just because we don’t need it (and so by Ockham’s Razor we don’t make it).