Miller–Reiner conjecture on Smith normal forms of differential-poset operators

Let $P$ be a differential poset, let $n\geq 0$, and let $[DU_n]$ be the matrix of the operator $DU_n$ with respect to the standard basis of ${\mathbb Z}P_n$. Write $p_n$ for the rank of $P_n$, and let $I_{p_n}$ be the $p_n\times p_n$ identity matrix. A Smith normal form over a ring is a diagonalization by invertible row and column operations whose nonzero diagonal entries form a divisibility chain.

Miller–Reiner conjecture. For any differential poset $P$ and any $n\geq 0$, the matrix

$$[DU_n]+xI_{p_n}$$

has Smith normal form over ${\mathbb Z}[x]$.

Smith normal form over ${\mathbb Z}[x]$ is not guaranteed for arbitrary matrices because ${\mathbb Z}[x]$ is not a principal ideal domain. The conjecture asserts that these particular matrices nevertheless admit such a form; the supplied source gives no resolution status.