Chollet's permanental Oppenheim inequality conjecture

Let ${\cal H}_n$ be the cone of positive-semidefinite Hermitian matrices, and let $A\circ B$ denote the Hadamard (entrywise) product. Chollet's conjecture. For $A,B\in{\cal H}_n$,

$$\mathop{\rm per}(A\circ B)\leqslant(\mathop{\rm per} A)(\mathop{\rm per} B).$$

The conjecture is presented as open; it is known in particular for $n\leqslant3$.