Cohen–Lenstra random-matrix conjecture over complete discrete valuation rings

Let $(R,\mathfrak{m})$ be a complete discrete valuation ring with residue field $R/\mathfrak{m}=\mathbb{F}_{q}$. Let $P_{1}(t),\ldots,P_{r}(t)\in R[t]$ be monic polynomials whose reductions $\overline{P}_{1}(t),\ldots,\overline{P}_{r}(t)\in\mathbb{F}_{q}[t]$ are distinct and irreducible. Fix finite-length $R$-modules $H_{1},\ldots,H_{r}$. For a finite-length $R$-module $H$, write $\lambda(H)$ for its partition type, and let $w(q,\lambda(H))=|\operatorname{Aut}_{R}(H)|$. Cohen–Lenstra random-matrix conjecture. One has

$$\lim_{n\rightarrow\infty}\operatorname{Prob}_{A\in\operatorname{Mat}_{n}(R)}\left(\begin{array}{c}\operatorname{coker}(P_{j}(A))\simeq H_{j} \\ \text{for }1\leq j\leq r\end{array}\right)=\prod_{j=1}^{r}\frac{1}{w(q^{\deg(P_{j})},\lambda(H_{j}))}\prod_{i=1}^{\infty}(1-q^{-i\deg(P_{j})}).$$

This conjecture extends Cohen–Lenstra-type distributions to cokernels of polynomial evaluations on uniformly random matrices over complete discrete valuation rings. The paper states that it resolves special cases in later theorems, but the general assertion remains open in the supplied text.