Positivity conjecture for matrix counts of a complementary diagonal permutation

For a positive integer $n$, let $v=(2n-1)(2n)(2n-3)(2n-2)\cdots3412$ be the indicated permutation, and let $I_v$ be its diagram. Write $M_{2n}(\overline{I_v},q)$ for the matrix count of rank-$2n$ matrices over $\mathbf{F}_q$ supported in the complement of $I_v$. Positivity conjecture. For $v=(2n-1)(2n)(2n-3)(2n-2)\cdots3412$, $M_{2n}(\overline{I_v},q)$ belongs to $\mathbb{N}[q]$. Computations in the source verify this for $n\leq40$, but the general assertion is left as a conjecture.