Cheong–Huang–Kaplan conjecture on cokernels of polynomial functions of random matrices

Fix finitely many monic polynomials $P_{1}(t),\dots,P_{l}(t)\in\mathbb{Z}_{p}[t]$ whose images in $\mathbb{F}_{p}[t]$ modulo $p$ are distinct and irreducible. For $1\leq j\leq l$, let $G_{j}$ be a finite module over $\mathbb{Z}_{p}[t]/(P_{j}(t))$. For $\bar{X}\in\operatorname{Mat}_{n}(\mathbb{F}_{p})$ satisfying $\operatorname{cok}(P_{j}(\bar{X}))\simeq G_{j}/pG_{j}$ for $1\leq j\leq l$, consider lifts $X\in\operatorname{Mat}_{n}(\mathbb{Z}_{p})$. Cheong–Huang–Kaplan's conjecture. The probability that $\operatorname{cok}(P_{j}(X))\simeq G_{j}$ for $1\leq j\leq l$ and $X\equiv\bar{X}\pmod p$ is

$$\underset{X\in\operatorname{Mat}_{n}(\mathbb{Z}_{p})}{\operatorname{Prob}}\left(\begin{array}{c} \operatorname{cok}(P_{j}(X))\simeq G_{j} \\ \text{for }1\leq j\leq l \\ \text{and } X\equiv\bar{X}\pmod p \end{array}\right)=p^{-n^{2}}\prod_{j=1}^{l}\frac{p^{\deg(P_{j})(\dim_{\mathbb{F}_{p}[t]/(P_{j}(t))}(G_{j}/pG_{j}))^{2}}}{|\operatorname{Aut}_{\mathbb{Z}_{p}[t]/(P_{j}(t))}(G_{j})|}\prod_{i=1}^{\dim_{\mathbb{F}_{p}[t]/(P_{j}(t))}(G_{j}/pG_{j})}(1-p^{-i\deg(P_{j})})^{2}.$$

The conjecture extends the paper's main theorem and is described as work in progress by Cheong, Huang, and Kaplan. It is intended as a non-asymptotic formula for joint cokernel distributions of polynomial expressions in random matrices, generalizing earlier results on random integral matrices.