Conjecture on the joint cokernel distribution of shifted random $p$-adic matrices

Let $X_m:=\{x_1,\ldots,x_m\}$ be a finite ordered subset of $\mathbb{Z}_p$ whose elements have distinct reductions modulo $p$, and let $\mathcal{C}_{X_m}:=\mathcal{C}(t+px_1,\ldots,t+px_m)$. For each $i$, write

$$H_i\cong\mathbb{Z}_p^{d_{\infty,i}}\times\prod_{r=1}^{\infty}(\mathbb{Z}/p^r\mathbb{Z})^{d_{r,i}},$$

and set $D_r:=\sum_{i=1}^m d_{r,i}$ and $s_i:=\operatorname{rank}_{\mathbb{F}_p}(H_i/pH_i)=\sum_{r=1}^{\infty}d_{r,i}+d_{\infty,i}$. Joint cokernel distribution conjecture. For every integer $m\geq 1$, one has $(H_1,\ldots,H_m)\in\mathcal{C}_{X_m}$ if and only if $s_1=\cdots=s_m$ and

$$\sum_{k=1}^{r-1}\left(\sum_{l=1}^{k}d_{l,i_k}\right)+(m-r)\sum_{l=1}^{r}d_{l,i_r}\leq D_1+\cdots+D_r$$

for every $1\leq r\leq m-2$ and $1\leq i_1,\ldots,i_r\leq m$. This conjecture proposes a complete characterization of which tuples of finitely generated $\mathbb{Z}_p$-modules occur as the cokernels of the shifted matrices.