Hole-one-descent conjecture for generalized monotone triangles

Let $A_{n,j}$ denote the number of alternating-sign matrices of size $n$ whose unique $1$ in the first row is in column $j$, and let $n\geq1$ and $i=1,\ldots,n-1$. Hole-one-descent conjecture. The identity

$$\alpha(n+1;1,2,\ldots,i-1,i+1,i,i+1,\ldots,n)=-\sum_{j=1}^{n}(j-i)A_{n,j}$$

should hold. The source proves the case $i=1$ and obtains part of the general decomposition, but explicitly says that the remaining contribution from two structural classes is an open problem.