Quantum equilibrium equivariance ($\rho (t_0)=|\psi (t_0){|}^2⟹\rho (t)=|\psi (t){|}^2$)

Both the ensemble position density transported by the guidance equation and the squared wave amplitude satisfy the same probability continuity equation. Subject to existence and uniqueness of that transport, an initial Born rule distribution stays a Born rule distribution. This is a preservation statement, not a derivation that every initial ensemble is already in equilibrium.