Disjoint classical purifications conjecture

Let $\psi_a$ and $\psi_b$ be two pure states of a generic statistical system, such as a quantum system. Let $F^{-1}(\{\psi_a\})$ and $F^{-1}(\{\psi_b\})$ be the sets of classical statistical states reproducing those states, and let $A$ and $B$ be the sets of pure classical states into which these preimage sets can be purified.

Disjoint classical purifications conjecture. The sets $A$ and $B$ have neither pure nor mixed states in common.

This extends the preceding convex-geometric restriction to systems with a continuum of extreme points and possibly infinite dimensions. The source gives no resolution and presents the statement among its physical conjectures.