Mészáros's critical-distribution conjecture for band-matrix cokernels

Let $p$ be prime. For each $t\in\mathbb{R}_+$, let $\nu_t$ be a distribution on the set of finite abelian $p$-groups. Let $n_1<n_2<\dots$ be positive integers, and let $B_i$ be a Haar-uniform $n_i\times n_i$ band matrix over $\mathbb{Z}_p$ with band width $w_i$. Assume that

$$\lim_{i\to\infty}p^{-w_i}n_i=t.$$

Mészáros's critical-distribution conjecture. For every finite abelian $p$-group $G$,

$$\lim_{i\to\infty}\mathbb{P}\bigl(\operatorname{cok}(B_i)\cong G\bigr)=\nu_t(G).$$

This conjecture predicts a one-parameter family of limiting cokernel distributions at the critical scale separating Cohen–Lenstra and non-Cohen–Lenstra behavior. The source states that the conjecture is still open.