Positivity conjecture for matrix counts of 123-avoiding permutations

Let $w\in\mathfrak{S}_n$ be a $123$-avoiding permutation, meaning that it contains no subsequence order-isomorphic to $123$, and let $I_w$ be its diagram. Write $M_n(\overline{I_w},q)$ for the matrix count of rank-$n$ matrices over $\mathbf{F}_q$ supported in the complement of $I_w$. Positivity conjecture. For every $123$-avoiding permutation $w\in\mathfrak{S}_n$, $M_n(\overline{I_w},q)$ belongs to $\mathbb{N}[q]$. This strengthens the preceding conjecture for the particular permutation $v$; computations reported in the source support it for the tested cases, while the general statement remains open.