The extremal permanent conjecture for unitarily invariant norms

Let $n$ be a positive integer, let $S(n,k)$ denote the class of matrices under consideration, let $G(n,k)$ be the distinguished matrix, and let ${\mathcal T}$ be the class of unitarily invariant matrix norms. For $T\in {\mathcal T}$, define

$$\tilde{G}(n,k)=\frac{G(n,k)}{T(G(n,k))}.$$

For $k\in\{2,\dots,n\}$ and $M\in S(n,k)$ satisfying $T(M)=1$, the permanent of $M$ is at most the permanent of $\tilde{G}(n,k)$.

Extremal permanent conjecture. For all $M\in S(n,k)$ such that $T(M)=1$,

$$\operatorname{Per}(M)\leq \operatorname{Per}(\tilde{G}(n,k)).$$

The bound is asserted to be tight because $\tilde{G}(n,k)\in S(n,k)$ attains it. The surrounding text gives a theorem for this extremal bound and, for the Frobenius normalization, identifies the global upper bound as $1/32$, attained at $\tilde{G}(4,2)$; the parser provides no evidence that this conjecture has been resolved independently.