Einstein stated that if the Michelson-Morley experiment was wrong then relativity is wrong, and isn't this what mainstream physics originates from? It seems pretty counter-productive, and if we receive a null answer then cool; however, if we receive a definitive number larger than zero, how do you think this experiment would shape the world afterward? I know this sounds like a conspiracy but I would like for a professional to shed some light on this subject.
Mainstream physics does not originate from the Michelson Morley experiment. Special relativity was introduced to solve a variety of problems, only one of which was the counterintuitive result of the Michelson Morley experiment (another is the apparent inconsistency of Maxwell's electromagnetism and Galilean physics).
Now, relativity solves these problems mathematically, but even this is not enough. The true power of a physical theory lies in its ability to make predictions. Special relativity (along with General relativity, which is an extension of it) makes a great deal of predictions (time dilation, length contraction, gravitational lensing, black holes, gravitational waves, an exact calculation of the procession of Mercury's orbit, etc.). Nearly all of these predictions have been verified experimentally to incredible precision. This is the true test of a theory: whether it holds up to experiment.
If in the future someone performed a Michelson Morley type experiment on another celestial body (which hasn't been done because, to be honest, it would be expensive and we're pretty sure we know what the outcome is) and the result turned out different, two things would happen:
The results would be met with extreme skepticism. This would be much like the "faster than light neutrinos" results from 2011. Since Einstein's theories have been experimentally verified so many times, this one experiment would be subject to a lot of questions. One would have to double, triple, and quadruple check the results. Furthermore, other researchers would have to conduct independent experiments to verify the results. This is one of the basic tenets of science: experimental results must be reproduced independently before taken seriously.
If, after all of this, the experiments kept giving anti-relativity results, and those results were consistent with each other in independent experiments, physics would be turned on its head. I personally can't think of a single mechanism that could cause such a result and still be compatible with the thousands of verified results predicted by relativity. Physicists around the world would be in a rush to try to explain this.
Personally, if these results were obtained and they were reproduced independently, the scientific community would be extremely excited. There would be no attempt to cover it up. To the contrary, physicists would simply try to find a new theory which can explain both the observed effects of Special and General Relativity while also predicting the new experiments. One thing is for sure, though: the new theory would not disprove relativity, in the same way that relativity doesn't disprove Newtonian mechanics. The new theory would simply extend what we already know to also include other phenomena.
I hope this helps!