One way we might see a violation of Bohm’s rule in quantum mechanics, a triple slit experiment:
The triple slit experiment uses three small openings instead of two. While it seems like a trivial change, if done correctly it allows for secondary interactions that could in principle violate the Born rule. Basically, if you just square the total wavefunction of the three slits, you get one probability distribution. If you calculate the secondary interactions you get a different distribution. The difference is extremely small, but in 2010 the experiment was performed, and found the Born rule held within experimental limits.
While this would seem to confirm the Born rule, the precision of the experiment was only to 1 part in 100, which isn’t very high. Unfortunately, even getting that level of precision is difficult with the triple-slit experiment. The Experimental Quest To Disprove A Quantum Postulate
I think this is not the same kind of deviation as we were discussing re: the pilot wave theory (which has to do more with the initial unknown positions of the particles in that theory, and less with the experimental setup), but it is interesting to see that this experiment is being performed. Always good to see the scientists are testing their basic postulates!
Not in the sense that the above paper describes, no. Most all pilot wave theories that I have read of i) keep the same Schrodinger equation as the corresponding standard quantum theory, and ii) postulate or derive the Born rule as a nigh-certainty (similar to the way that the 2nd law of thermodynamics comes about from statistical mechanics). It is the tiny amount of wiggle room left open in that “nigh-certainty” which allows for the possibility of violations of the Born rule in pilot-wave theories.