As Orzel reported:
That's what the NJP paper linked above is about. One of the ways you might get the Born rule from some deeper principle would be to have it be merely an approximation to some more fundamental structure. That, in turn, might very well involve a procedure other than "squaring" the wavefunction to get the probability of various measurement outcomes. In which case, you would expect to see some higher-order contributions to the probability-- the wavefunction cubed, say, or to the fourth power.
Sadly, for fans of variant models of quantum probability, what they actually do is the latter. They don't see any deviation from the ordinary Born rule, and can say with confidence that all the higher-order contributions are zero, to something like a hundredth of a percent.
Of course, this won't stop the continuation of the search, because that is what we do. But it is amazing that QM has withstood numerous challenges throughout its history.