Real-part largest-eigenvalue conjecture for the permanent matrix

Let $\mathcal{H}_n$ be the set of $n\times n$ positive semidefinite Hermitian matrices. For $A=[a_{i,j}]\in\mathcal{H}_n$, let $A(i,j)$ be obtained by deleting row $i$ and column $j$, and define $F_A=[f_{i,j}]$ by

$$f_{i,j}=a_{i,j}\operatorname{per}(A(i,j)).$$

Let $\lambda_{\max}^{\mathbb{R}}(F_A)$ denote the maximum eigenvalue of the symmetric part of $F_A$ over $\mathbb{R}^{n\times n}$, formed by taking the real part of each entry of $F_A$. Real-part eigenvalue conjecture. For every $A\in\mathcal{H}_n$,

$$\lambda_{\max}^{\mathbb{R}}(F_A)\leq\operatorname{per}(A).$$

The supplied text does not give evidence resolving this newly introduced conjecture, so its database status remains open.