No Signalling and Unknowable Bohmian Particle Positions

We exhibit in a model with simple dynamics, specifically a particle in a square box or two particles in one dimensional boxes, that if an experimenter can prepare the initial wave function of a system, the maximal information about the positions of Bohmian particles that is compatible with "no signalling" is that they are distributed according to $|\psi(x)|^2$. In particular, the positions cannot be prepared independently from the wave function. Any sharper "actual" position of the particle must be inaccessible since it could be used to send signals instantaneously. This is a consequence of the non-local character of the Bohmian dynamical law.