Central-rank divisibility conjecture for monotonic-triangle rank polynomials

Let $r_{{\cal{MN}}_n}(t)$ be the rank-generating polynomial of the set ${\cal{MN}}_n$ of monotonic triangles, and let $\operatorname{ASM}(n)$ denote the number of alternating sign matrices of size $n$. Central-rank divisibility conjecture. The coefficient at degree $\left[{n+1\choose3}/2\right]$ satisfies

$$\operatorname{Coeff}_{\left[{n+1\choose3}/2\right]}r_{{\cal{MN}}_n}(t)\mid\operatorname{ASM}(n).$$

The conjecture predicts an arithmetic relation between the central rank coefficient of the monotonic-triangle rank polynomial and alternating sign matrix enumeration; the source gives no resolution.