I'm reviving and expanding this question, because of the new paper today, by Aaronson et al. The more general question is:
How does quantum-potential Bohmian mechanics relate to no-go theorems for psi-epistemic theories?
According to Bohmian mechanics, there is a wavefunction as in conventional quantum mechanics, which evolves according to a Schrödinger equation, and then a set of classical degrees of freedom, which evolve according to the gradient of the phase of this wavefunction. A Born-rule probability distribution for the classical degrees of freedom is preserved by this law of motion, so Bohmian mechanics provides a deterministic hidden-variables theory, albeit a nonlocal one.
However, the equations of motion for the classical variables can be rewritten in terms of two potentials, the local potential appearing in the Schrödinger equation, and a nonlocal "quantum potential" which depends on the form of the wavefunction. This means that Bohmian mechanics for any given wavefunction can be replaced by a nonlocal deterministic theory in which there is no wavefunction. (Note that the quantum potential will be time-dependent unless the wavefunction was an energy eigenstate; thanks to Tatiana Seletskaia for emphasizing this point.)
Meanwhile, there has recently been theoretical interest in ruling out so-called "psi-epistemic" hidden-variables theories, in which the quantum wavefunction is not part of the ontology. Obviously, "quantum-potential Bohmian mechanics" is a psi-epistemic theory. It ought to be relevant to this enterprise. But it may require some mental gymnastics to bring it into contact with the PBR-like theorems, because the mapping from quantum mechanics to these quantum-potential theories is one-to-many: scenarios with the same quantum equations of motion, but different initial conditions for the wavefunction, correspond to different equations of motion in a quantum-potential theory.
NOTE 1: Empirically we only have one world to account for, so a real-life quantum-potential theorist in principle shouldn't even need their formalism to match the whole of some quantum theory. Quantum cosmology only needs one wavefunction of the universe, and so such a theorist would only need to consider one "cosmic quantum potential" in order to define their theory.
NOTE 2: Here is the original version of this question, that I asked in January:
The "PBR theorem" (Pusey-Barrett-Rudolph) purports to show that you can't reproduce the predictions of quantum mechanics without supposing that wavefunctions are real. But it always seemed obvious to me that this was wrong, because you can rewrite Bohmian mechanics so that there's no "pilot wave" - just rewrite the equation of motion for the Bohmian particles, so that the influence coming from the pilot wave is reproduced by a nonlocal potential. Can someone explain how it is that PBR's deduction overlooks this possibility?