Given a matrix $A \in M_d(\mathbb{Z}_p)$ (with nonzero determinant), viewed as a map $\mathbb{Z}_p^d \to \mathbb{Z}_p^d$, I am interested in the sequence of abelian $p$-groups $\{coker(A^n)\}_{n \geq 1}$ (equivalently, invariant factors/Smith normal forms of all powers). For certain matrices, it is quite clear that this sequence of groups follows a pattern, e.g. if $A$ is diagonal we only need to know the cokernel of $A$ and this determines all the others. In general, I am hoping there is some nice, finite combinatorial object that packages all of the information of this sequence, but any way to specify any such sequence via, say, a finite number of integers would be a very good start.

Note: Someone asking more or less the same question but over any PID made a stronger conjecture which people in the comments were optimistic could be proven, but this proof never appeared. Such a proof would immediately solve my problem.