The converse-type conjecture for sparse Haar-random matrices

For each $n$, let $\Sigma_n=(\sigma_{n,1},\ldots,\sigma_{n,n})$, where each $\sigma_{n,j}$ is a subset of $[n]=\{1,2,\ldots,n\}$. Let $X_n\in\operatorname{M}_n(\mathbb{Z}_p)$ be Haar-random on this support: $(X_n)_{i,j}=0$ for $i\notin\sigma_{n,j}$, while the entries with $i\in\sigma_{n,j}$ are independent Haar-random elements of $\mathbb{Z}_p$. Define

$$|\Sigma_n|:=\sum_{i=1}^n|\sigma_{n,i}|,$$

and write $\operatorname{cok}(X_n)$ converges to CL when its distribution converges to the Cohen–Lenstra distribution as $n\to\infty$. Converse-type conjecture. For every sequence $(a_n)_{n\geq 1}$ satisfying $n\leq a_n\leq n^2$ and

$$\lim_{n\to\infty}\left(\frac{a_n}{n}-\log_p n\right)=\infty,$$

there is a sequence $(\Sigma_n)_{n\geq1}$ such that $\operatorname{cok}(X_n)$ converges to CL and $|\Sigma_n|=a_n$ for all $n$.

The conjecture is presented as a converse-type result to the necessary lower bound for convergence to the Cohen–Lenstra distribution, and is described as best possible with respect to that bound. No resolution evidence is supplied in the source.