Poisson distribution conjecture for cokernels of random matrices over p-adic integers

Let $\mathbb{Z}_{p}$ be the ring of $p$-adic integers. Fix distinct $d_{1},\dots,d_{r}\in\mathbb{Z}_{\geq 1}$ and not necessarily distinct $k_{1},\dots,k_{r}\in\mathbb{Z}_{\geq 0}$. For each $d$, let $\mathcal{M}(d,p)$ be the set of monic irreducible polynomials of degree $d$ in $\mathbb{F}_{p}[t]$. For $(\bar{P}_{1},\dots,\bar{P}_{r})\in\prod_{j=1}^{r}\mathcal{M}(p,d_{j})$, let $P_{j}$ be any lift of $\bar{P}_{j}$ to $\mathbb{Z}_{p}[t]$, and let $A$ be Haar-random in $\operatorname{Mat}_{n}(\mathbb{Z}_{p})$. Poisson distribution conjecture. The limit

$$\lim_{p\to\infty}\operatorname{Prob}_{A\in\operatorname{Mat}_{n}(\mathbb{Z}_{p})}\left(\begin{array}{c}|\operatorname{coker}(P_{j}(A))|=p^{d_{j}k_{j}}\text{for all }(\bar{P}_{1},\dots,\bar{P}_{r})\in\prod_{j=1}^{r}\mathcal{M}(p,d_{j})\end{array}\right)$$

must exist, and

$$\lim_{n\to\infty}\lim_{p\to\infty}\operatorname{Prob}_{A\in\operatorname{Mat}_{n}(\mathbb{Z}_{p})}\left(\begin{array}{c}|\operatorname{coker}(P_{j}(A))|=p^{d_{j}k_{j}}\text{for all }(\bar{P}_{1},\dots,\bar{P}_{r})\in\prod_{j=1}^{r}\mathcal{M}(p,d_{j})\end{array}\right)=\prod_{j=1}^{r}\frac{e^{-1/d_{j}}(1/d_{j})^{k_{j}}}{k_{j}!}.$$

This conjecture generalizes the preceding theorem, whose limiting probability for all the cokernels to be trivial is $\prod_{j=1}^{r}e^{-1/d_{j}}$. The claimed formula is the joint distribution of independent Poisson random variables with means $1/d_{1},\dots,1/d_{r}$. The authors report little empirical evidence and invite further examples supporting or disproving it, so its status remains open.