Frozen-corner determinant conjecture for alternating sign matrices

Let $A_n$ be the number of $n\times n$ alternating sign matrices, and let $B_{n,s}$ be the number of such matrices having an $s\times s$ square of zero entries located in a corner. Define

$$g_n(z):={}_2F_1\left(\left.\genfrac{}{}{0pt}{}{ -n+1\quad n }{ 2n } \right| 1-z\right),$$ $$f_i^{\pm}(z):=\left[1\pm(-1)^i z\right]\frac{(1-z)^{i-1}}{z^i}g_{n-s+i}(z).$$

For fixed $n$ and $s$, let $M$ be the $s\times s$ matrix with entries

$$M_{ij}:=\frac{A_{n-s+j}}{A_{n-s+j-1}}\oint_{C_0}\oint_{C_0}\frac{f_i^+(z)f_j^-(w)}{1-z-w}\frac{\mathrm{d}z\,\mathrm{d}w}{(2\pi\mathrm{i})^2},\qquad i,j=1,\dots,s,$$

where $C_0$ is a small closed contour around the origin containing no other singularity of the integrand.

Frozen-corner determinant conjecture. The number $B_{n,s}$ may be evaluated as

$$B_{n,s}=A_n\det_{1\leq i,j\leq s}(1-M),$$

where $M$ is equivalently given by the explicit finite-sum expression in the source or by the preceding hypergeometric and contour-integral definitions.

The formula is proposed on the basis of numerical support and is intended to enumerate ASMs with a prescribed frozen corner. Its status is unresolved in the supplied source.