Chollet's permanent conjecture for positive semidefinite matrices

Let $A,B e 0$ be Hermitian positive semidefinite matrices of the same size, and let $A\circ B$ denote their Hadamard product. Chollet's conjecture.

$$\operatorname{per}(A\circ B)\leq \operatorname{per}(A)\,\operatorname{per}(B).$$

The conjecture is a permanent analogue of determinant inequalities for Hadamard products of positive semidefinite matrices. For $n\geq 5$, it remains open.