Cheong and Kaplan's polynomial push-forward cokernel conjecture

Let $P(t) \in \mathbb{Z}_p[t]$ be a non-constant monic square-free polynomial, written as

$$P(t)=P_1(t)\cdots P_l(t),$$

where each $P_j(t)\in\mathbb{Z}_p[t]$ is monic and its reduction $\bar{P}_j(t)$ modulo $p$ is distinct and irreducible in $\mathbb{F}_p[t]$. For a finite $\mathbb{Z}_p[t]/(P(t))$-module $G$, let $A_n\in\mathrm{M}_n(\mathbb{F}_p)$ satisfy $\operatorname{cok}(\bar{P}(A_n))\simeq_{\mathbb{F}_p[t]}G/pG$. Write $q_j=p^{\deg(P_j)}$, let $\mathbb{F}_{q_j}=\mathbb{F}_p[t]/(\bar{P}_j(t))$, and set $r_{q_j}(G)=\dim_{\mathbb{F}_{q_j}}(G/pG\otimes_{\mathbb{F}_p[t]}\mathbb{F}_{q_j})$. Cheong and Kaplan's conjecture. One must have

$$\operatorname{Prob}_{X\in\mathrm{M}_n(\mathbb{Z}_p)^{\mathrm{Haar}}}\left(\operatorname{cok}(P(X))\simeq_{\mathbb{Z}_p[t]}G\mid X\equiv A_n\pmod p\right)=\frac{1}{|\operatorname{Aut}_{\mathbb{Z}_p[t]}(G)|}\prod_{j=1}^{l}p^{r_{q_j}(G)^2}\prod_{i=1}^{r_{q_j}(G)}(1-q_j^{-i})^2.$$

This conjecture generalizes results of Friedman and Washington and subsequent special cases proved by the authors, Cheong and Kaplan, and Cheong, Liang, and Strand; it predicts the conditional distribution of cokernels of polynomial images of random integral matrices when the polynomial has distinct irreducible residue factors.