Miller–Reiner Smith normal form conjecture for differential posets

Let $P$ be a differential poset, let $p_n$ denote the size of its rank-$n$ set, and let $U_{n-1}D_n$ be the composite of the down operator $D_n$ from rank $n$ to rank $n-1$ with the up operator $U_{n-1}$ from rank $n-1$ to rank $n$. The map is considered over the polynomial ring $\mathbb{Z}[t]$. Miller–Reiner's conjecture. For all differential posets $P$ and all $n$, the map

$$U_{n-1}D_n+tI:\mathbb{Z}[t]^{p_n}\to\mathbb{Z}[t]^{p_n}$$

has a Smith normal form over $\mathbb{Z}[t]$. Since $\mathbb{Z}[t]$ is not a principal ideal domain, this would extend the usual Smith-normal-form theory beyond the PID setting and provide structural information about the associated cokernels.