Higher d-separation obstructs toric vanishing ideals

Let $G$ be a directed acyclic graph (DAG), and let $i,j$ be vertices of $G$. A d-separating set for $i$ and $j$ is a set of vertices that d-separates them; consider the minimum cardinality of such a set. Higher d-separation conjecture. If the minimum size of a d-separating set for $i$ and $j$ is at least $2$, then $I_G$ is not toric. This conjecture would exclude many DAGs from having toric vanishing ideals and is intended as an auxiliary tool for the classification results; it remains open in the supplied text.