The standard stuff of the Copenhagen interpretation is derivable from Bohmian mechanics, to the extent that the collapse in Copenhagen is sufficiently well-specified. This obviously prevents finding experimental disagreements. One thing that is somewhat problematic from an orthodox perspective is that of arrival times since, in some sense, nothing is arriving. Time is problematic, in general, in standard QM.

One recent paper that explores Bohmian mechanical predictions concerning arrival times is https://arxiv.org/abs/1802.07141 It explains how it is not clear in the standard formulation what the answer should be and then provides the Bohmian answer since arrival times do exist as there are particles with trajectories that go somewhere. I suspect that if the predictions were proven right, someone would look at standard QM and deduce that that should have been considered the right answer all along.

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As for quantum field theory and its divergences, while the work has only been done on some of the simpler models, there is a process called "Interior Boundary Conditions" which removes the need for ultraviolet cutoffs: https://arxiv.org/abs/1703.04476

The basic idea is that the free Hamiltonian's natural domain is not appropriate when adding in the particle creation/annihilation operators and thus a boundary condition representing those interactions needs to be chosen. More or less, the appropriate set of wave functions are those whose probabilities adjust themselves appropriately for the particle creation and annihilation. This approach was inspired by a Bohmian perspective, but it is a mathematical methodology that does not require accepting that perspective.

Thus, there is some hope that the divergences are solvable already with what we know.

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As far as I know, the most fundamental issue of relativity and quantum mechanics is that of time. Namely, there is a strong notion of "now" seemingly from QM and there is a strong notion of "not now" from relativity. Neither statements are absolute, but the tendencies are there. Bell's work suggests that nature is either nonlocal (nowish) or there is something not quite defined about experiments from what we think there is (many universes, experimental results not finalized until actually compared, somehow). In Bohmian mechanics, the issue of "now" is quite acute as the trajectories of each particle depends not only on the wave function, but on the position of all the other particles "now". This is problematic as there is no definite "now" in relativity.

One solution is to define a "now", namely, define slices of space-time which define the instants of now. This can be done in a Lorentz covariant way in flat space-time in such a manner that we cannot detect the foliation, by defining the foliation from the dynamic structures already present: https://arxiv.org/abs/1307.1714 The authors are not necessarily advocating for this as it seems philosophically wrong, but, again, Bell's work strongly suggests something of this kind needs to be true, no matter how distasteful it is.

This would necessarily imply that space-time can have a foliation, something which is not always true. Perhaps the dynamics defining the space-time metric in the quantum framework ensures that this does indeed stay the case.