Permanent inequality for rank-at-most-two matrices

Let $T$ be a matrix of size $n\times n$, and let $\operatorname{per}(T)$ denote its permanent. Form the $2n\times 2n$ block matrix with every block equal to $T$:

$$\begin{pmatrix}T&T\T&T\end{pmatrix}.$$

Rank-two permanent inequality. For any matrix $T$ with rank at most $2$,

$$\operatorname{per}\begin{pmatrix}T&T\T&T\end{pmatrix}\leq \binom{2n}{n}\operatorname{per}(T)^2.$$

The conjecture was proposed as a potentially useful tool in finite free probability. The source describes it as an unproved conjecture motivating the determinantal identity, and does not provide evidence of a resolution.