I judge by the fact that there aren't very many people doing experiments meant to distinguish the various proposals that physicists continue to generally believe there isn't much that can be done to distinguish the various interpretations from each other. Given the incentives to complete such an experiment and even just eliminating one possibility (even if one couldn't nail down which is true), and the fact that it seems to me many of the possible experiments that could do this are more lab-bench style experiments with a laser and a handful of mirrors and electronics rather than Large Hadron Collider-class experiments, I'm guessing the idea that the theories are distinguishable is a minority view, and one that the minority has not yet managed to do something so compelling with that the majority has no choice but to take note.