Toricity characterized by the shortest trek ideal

Let $G$ be a directed acyclic graph (DAG), with vanishing ideal $I_G$ and shortest trek ideal $ST_G$. Shortest-trek characterization conjecture. The vanishing ideal $I_G$ is toric if and only if

$$I_G=ST_G.$$

The equality identifies the toric structure through the shortest trek map. The paper notes that shortest trek ideals need not always equal vanishing ideals and presents this equivalence as an open main conjecture.