The extremal completion conjecture for bordered-permutation matrices

Let $E_m$ be the $m\times m$ $(0,-1)$-matrix with $-1$s in positions $(2,m),(3,m-1),\ldots,(m,2)$, and let $F_m$ be the $m\times m$ $(0,-1)$-matrix with $-1$s in positions $(1,m-1),(2,m-2),\ldots,(m-1,1)$. For a matrix $A$, write $A\oplus B$ for its block direct sum, and let $I_1$ denote the $1\times 1$ identity matrix. The extremal completion conjecture. If $n\geq 4$ is even, then

$$E_{n/2}\oplus F_{n/2}$$

is the $n\times n$ bordered-permutation $(0,-1)$-matrix with the largest number of alternating sign matrix completions. If $n\geq 5$ is odd, then

$$E_{(n-1)/2}\oplus(-I_1)\oplus F_{(n-1)/2}$$

is the $n\times n$ bordered-permutation $(0,-1)$-matrix with the largest number of alternating sign matrix completions. The conjecture proposes an extremal structure for completions of bordered-permutation matrices; the supplied text gives no resolution, so its status remains open.