Prime-diagonal rank conjecture for graphical Hermite simplices

Let $G$ be the topological ordering of a graph on $[n]$, and let $B$ be the matrix over ${\mathbb{Z}}/p{\mathbb{Z}}$ given by

$$B=\sum_{(i,j)\in G}E_{i,j}.$$

Let $r_p$ be the rank of $B$ over ${\mathbb{Z}}/p{\mathbb{Z}}$, and let $\alpha_i$ be the elementary divisors of $A_{\vec p,G}$. Prime-diagonal rank conjecture. The elementary divisors satisfy $\alpha_{r_p}=1$ and $\alpha_{r_p+1}>1$.

This conjecture makes precise the experimental observation that, for a prime diagonal, the rank of the adjacency matrix over ${\mathbb{Z}}/p{\mathbb{Z}}$ determines how powers of $p$ are distributed among the nonunit elementary divisors, extending the preceding result from bipartite graphs to all graphs. Its status is not established in the supplied text.