The critical-window conjecture for cokernels of random band matrices

Let $p$ be a prime, let $(\nu_{\alpha})_{\alpha\in\mathbb{R}}$ be a proposed one-parameter family of distributions on the set of finite abelian $p$-groups, and let $n_1<n_2<\dots$ be positive integers. For each $i$, let $\mathbf{B}_i$ be an $n_i\times n_i$ band matrix over $\mathbb{Z}_p$ with band width $w_i$. Assume that

$$\lim_{i\to\infty} \bigl(w_i-\log_p(n_i)\bigr)=\alpha.$$

For every finite abelian $p$-group $G$, critical-window cokernel conjecture.

$$\lim_{i\to\infty}\mathbb{P}\bigl(\operatorname{cok}(\mathbf{B}_i)\cong G\bigr)=\nu_{\alpha}(G).$$

The conjecture predicts a family of limiting cokernel laws at the critical scale $w_i\sim\log_p(n_i)$. The paper's preceding tightness result motivates this formulation, but the supplied text gives no evidence that the conjecture has been resolved.