Total nonnegativity of the permanental analogue of the Sihong polynomial

Let $x$ be a matrix of indeterminates, and for $n\geq 1$ set

$$I=[2n]\smallsetminus 2\mathbb Z,\qquad J=[2n]\cap 2\mathbb Z.$$

For index sets $A$ and $B$, write $x_{A,B}$ for the corresponding submatrix and $\operatorname{per}(x_{A,B})$ for its permanent. A polynomial is totally nonnegative if it evaluates to a nonnegative real number on every totally positive matrix. The polynomial in question is

$$\operatorname{per}(x_{[n],[n]})\operatorname{per}(x_{[n+1,2n],[n+1,2n]})-\operatorname{per}(x_{I,I})\operatorname{per}(x_{J,J}).$$

Permanental Sihong polynomial conjecture. The polynomial above is totally nonnegative for all $n$. This would provide a permanental analogue of the known totally nonnegative determinant polynomial obtained by replacing the permanents with determinants and reversing the order of the two products. The claim is established in the source for small values of $n$, while its validity for all $n$ is left as a future direction.