  The trajectories of the pilot-wave formulation of quantum mechanics and hence
its empirical predictions may be recovered via the dynamics of a density
function on the configuration space of a system, without reference to a
physical wave function. We label such formulations `CSD frameworks.' But this
result only holds if a particular, apparently ad hoc condition, broadly
speaking equivalent to the single-valuedness of the wave function in standard
quantum mechanics, is imposed. Here we relax this condition. We describe the
types of scenarios in which this would lead to deviations from quantum
mechanics. Using computational models we ask how the degree of
`non-quantumness' of a state, suitably defined, changes with time. We find that
it remains constant in time even under non-trivial dynamics, and argue that
this implies that a dynamical justification of the Wallstrom condition is
unlikely to be successful. However, we also make certain observations about
stationary states in CSD frameworks, which may offer a way forward in
justifying the Wallstrom condition.
