Safe-gluing decomposition for chordal DAGs with toric vanishing ideals

Assume the higher d-separation conjecture: whenever two vertices of a DAG have minimum d-separating-set size at least $2$, the vanishing ideal is not toric. Let $G$ be a chordal directed acyclic graph (DAG) with toric vanishing ideal. A safe gluing combines two DAGs along an $N$-clique under the safe-gluing conditions. Chordal safe-gluing decomposition conjecture. There exist DAGs $G_1$ and $G_2$ with toric vanishing ideals such that $G$ can be obtained by safely gluing $G_1$ and $G_2$ at an $N$-clique. This conditional conjecture would provide a decomposition of chordal toric DAGs into smaller toric pieces; it remains open because it depends on the higher d-separation conjecture.