Loss of preparation information after a nondegenerate projective measurement

For an ideal projective measurement with rank-one projectors $P_m=|m⟩⟨m|$, every initial state that can give outcome $m$ leaves the same conditional state $|m⟩⟨m|$. Every subsequent procedure using only that system and the recorded outcome therefore has the same statistics for all such initial hypotheses. It cannot improve the posterior about the preparation. This claim concerns additional measurements after the first collapse, not access to fresh copies or a record preserved before the measurement.