As far as I understand – and Wikipedia seems to support this – pilot wave theory, or actually any hidden variable theory, or actually any interpretation of Quantum Mechanics, they are different mathematical models that produce exactly the same predictions for experimental outcomes.
You (and everyone who replied you so far) misunderstood: if you read the article, you'll see that this is not the issue.
This is about the reality of trajectories built-into the pilot wave theory (such a concept which doesn't even exist in "text-book" interpretation of quantum mechanics, which says particles don't have positions or trajectories until you measure them, upon which they "decide" their position through an "artificial" process called wavefunction collapse; notably, as of now, wavefunction collapse and the reality of wavefunction is still a subject of debate).
There was a paper "Surrealistic Bohm Trajectories" which argued that trajectories predicted by pilot wave theory don't make any sense as real trajectories in general, hence these "trajectories" cannot correspond to real trajectories.
A careful analysis which properly takes the nonlocality into account shows the flaw in their argument (arXiv:quant-ph/0010020, omitted in the text linked), and this is the corresponding experiment which confirms their analysis (using weak measurements on a pair of entangled photons to sketch "average" trajectories in a WWM setup).
If you like, this is about the interpretation of a concept within an interpretation.
While I agree that the title is a little bit flashy, experimental confirmation for the reality of Bohmian trajectories is an important topic.
You're 100% correct - this is merely a direct presentation of the weirdness that results if you choose to interpret this experiment in the language of pilot wave theory. The physical results are the same as you would predict using normal methods, and as such it's a real stretch to consider this experimental support for Bohm's interpretation.
One thing I've never been clear on is whether this is empirically the case, or theoretically the case. That is to say, is it theoretically possible that someday we may find evidence that supports Bohm's interpretation and falsifies the others? Or are they truly mathematically equivalent theories?
If memory serves, as far as classical quantum mechanics goes, Bohm's interpretation is 100% mathematically equivalent to the standard interpretation, making it observationally impossible to distinguish the two (much like many worlds vs. Copenhagen). It would be in extensions to the theory where things might differ, though I don't know much about attempts to extend pilot wave theory to QFT.
> Bohm's interpretation is 100% mathematically equivalent to the standard interpretation, making it observationally impossible to distinguish the two
This is not quite correct.
It is mathematically equivalent to Copenhagen only at times of 'quantum equilibrium', which is when ρ = φ² (probability density is the square of the wavefunction). Copenhagenists call this the Born rule and treat it as axiomatic. Bohmians don't since they don't need to, since in Bohmian mechanics a system which is not in quantum equilibrium will evolve to be in quantum equilibrium in a very very short time.
Which gives a way to empirically distinguish pilot-wave theory if the universe began in quantum non-equilibrium. (We have no reason to think it did, but we have no reason to think it didn't, so, shrug). If it did, the early universe would have evolved differently in the first femtosecond after the big bang, which should theoretically be detectable to us by analysis of CMB anisotropies.
This has been worked on: a friend of mine did a masters project which among other things involved porting some simulation code on very-early-universe numeric simulations under quantum non-equilibrium (IIRC from fortran 77 to fortran 95, because physicists), under Mike Towler.
That's correct. If we knew of an experimental technique to distinguish interpretations of quantum mechanics we would be in a much better place than we are now. This article is a classic of article titles stretching science journalism, stretching the press release, stretching the paper, stretching the basic result.
There may be consequences of each different interpretation, that might leave it open to different results in some conditions. I remember Many World theory claiming there would be some unexplained signals in cosmic background radiation.
Iirc pilot wave theory better explain wave/particle duality but afaik there isn't yet a theoretical experiment I know of which could tell it apart from the current theory.
https://en.wikipedia.org/wiki/Pilot_wave
So how can, even in theory, any experiment support one interpretation any more or less than all the others?