Recursive classification of DAGs with toric vanishing ideals by gluing operations

Let $G$ be a directed acyclic graph (DAG) with toric vanishing ideal $I_G$. The relevant constructions are safe gluing, gluing at sinks, and adding a sink. Recursive classification conjecture. At least one of the following holds: $G$ is a complete DAG; $G$ is obtained by safe gluing or gluing at sinks of two smaller DAGs with toric vanishing ideals; or $G$ is obtained by adding a sink to a smaller DAG with a toric vanishing ideal. This is the paper's second main conjecture and remains open; it gives the explicit recursive form of the broader constructive classification claim.