Local-isometry target-state realization witness

Let $p$ be a finite Bell behavior and let $\psi$ be a bipartite target state. Suppose that a chosen finite tensor-product quantum realization reproduces the probability table of $p$ and has shared state $\rho$. If local isometries $V_A$ and $V_B$ satisfy \[ (V_A\otimes V_B)\rho(V_A\otimes V_B)^\dagger = \psi, \] then the chosen realization supplies a target-state realization witness for $p$ and $\psi$. The statement is existential in the chosen realization; it does not assert that every realization of $p$ extracts $\psi$.