Safe gluing preserves equality of the vanishing and shortest trek ideals

Let $G_1$ and $G_2$ be directed acyclic graphs (DAGs), and let $I_{G_i}$ be their vanishing ideals and $ST_{G_i}$ their shortest trek ideals. Suppose that $I_{G_1}=ST_{G_1}$ and $I_{G_2}=ST_{G_2}$. Let $G$ be obtained by safely gluing $G_1$ and $G_2$ at an $N$-clique. Safe-gluing equality conjecture. Then $I_G=ST_G$, and consequently $I_G$ is toric. This is a technical form of the safe-gluing conjecture; the paper proves partial and related preservation results, but the stated general claim remains open.