Cohen–Lenstra convergence criterion for general support patterns

Let $p$ be a prime, let $[?]$ be a sequence of support patterns, and let $X_n[?]$ be the Haar-random matrix over $[?]$ supported on $[?]$. Write $[?]$ for the reduction of $X_n$ modulo $p$, and let $[?]$ denote the Cohen–Lenstra distribution's mass at the trivial group. Then

General support-pattern conjecture.

$$\mathrm{cok}(X_n) \rightrightarrows \mathrm{CL} \quad\Longleftrightarrow\quad \lim_{n\to\infty}\mathbb{P}(\mathrm{det}(\overline{X_n})\ne 0)=\mathrm{cl}(0).$$

The preceding theorems establish this equivalence for several stair-shaped support families, including band matrices and matrices with two symmetric stair-shaped zero regions. The conjecture extends that criterion to arbitrary support patterns, while the source notes that analogous rank-$r$ criteria fail for $r\geq1$.