Experiments justify 'predictions' of certain mathematical models, not 'interpretations'. As for a belief in interpretations, it's more of a philsophy, and Feynman said just 'shut up and calculate'.
I agree that interpretations of QM are currently in the realm of philosophy, but (and, at the risk of being a fool in saying this) I disagree with Feynman when he says “shut up and calculate” [1].
I tend to lean towards Sean Carrol’s approach of the interpretation is important and we should spend real effort thinking about the different impacts of different interpretations. After all, Einstein didn’t get to general relativity just by calculating. It first required deep thought and consideration to decide what to calculate. And I’m not convinced that the interpretations will forever remain in the realm of philosophy (though, I accept I could be totally wrong about this).
[1] I don’t really disagree with Feynman on this. I think it’s perfect advice for people getting started in really learning QM (which is beyond where I am!). Being very familiar with the mechanics and being able to calculate fluently are probably a good starting point for people learning QM. But I do think that it’s worth experts in the field spending some time beyond the mathematical fundamentals.
If we just admit the possibility that the math was right all along we win, the math works, and we lose “I am a unique, distinct entity.”
I’m on team math.