Sharp distinction between expansive and abelian expansive networks

Let $q$ be an alphabet size, and let $E[D,q]$ denote the set of expansive networks on a digraph $D$ over an alphabet of size $q$, while $EA[D,q]$ denotes the set of abelian expansive networks on $D$ over that alphabet. Sharp distinction conjecture. For all $q$, there exists a digraph $D$ such that

$$E[D,q] \ne \emptyset \quad\text{but}\quad EA[D,q] = \emptyset.$$

The conjecture asserts that expansive networks can exist on some digraphs even when no abelian expansive network exists, despite the proved bounds for complete graphs and the corresponding polynomial bound for linear networks. Whether such a distinction holds for every alphabet size remains open.