Equality of Krull-dimension bounds for Khovanov-Rozansky homology

Let $G$ be a directed graph and let $E$ be a subset of its edge set $E(G)$. Write $\mathscr{H}_0^\mathbb{Q}(G)$ for the degree-zero Khovanov-Rozansky homology over $\mathbb{Q}$ and $\mathscr{H}_0(G)$ for the corresponding homology over $\mathbb{Z}$. For a module over a ring, write $\mathrm{K}\text{-}\dim$ for its Krull dimension. Krull-dimension equality conjecture. For every such $G$ and $E$,

$$\mathrm{K}\text{-}\dim_{\mathbb{Q}[E]} \mathscr{H}_0^\mathbb{Q}(G)=\mathrm{K}\text{-}\dim_{\mathbb{Z}[E]} \mathscr{H}_0(G)-1.$$

The conjecture asserts that the inequality proved in the surrounding discussion is always an equality; establishing this would rule out the possibility that rational coefficients yield a strictly stronger upper bound for the graph invariant $\alpha_E(G)$.