Suppose $A$ is a positive semi-definite operator on $V \otimes W$ such that $\operatorname{tr}(A) > 0$, where $V$ and $W$ are finite-dimensional complex vector spaces. Let $P$ be an orthogonal projection on $V \otimes W$, and let $Q$ and $R$ be orthogonal projections on $V$ and $W$, respectively. If we have $$PAP = (Q \otimes R) A (Q \otimes R) ,$$ does $P$ necessarily have the form of a tensor product $P = S \otimes T$ for some $S$ and $T$?
In the special case that $A$ is positive definite, $P = Q \otimes R$. I am interested in any similar statement that can be made under weaker conditions on the positivity of $A$.