Marcus's block permanent conjecture

Let $A\in{\cal H}_{mk}$ be partitioned into $k\times k$ blocks $A_{i,j}$, with $i,j=1,2,\ldots,m$. Let $G$ be the $m\times m$ matrix with entries

$$G_{i,j}=\mathop{\rm per}(A_{i,j}).$$

Marcus's block permanent conjecture. Then

$$\mathop{\rm per} A\geqslant \mathop{\rm per} G.$$

If every diagonal block $A_{i,i}$ is positive definite, equality holds if and only if

$$A=A_{11}\oplus A_{22}\oplus\cdots\oplus A_{mm}.$$

The source records the $m=2$ case and a real-matrix result, but leaves the general conjecture unresolved.