Minimality conjecture for generators of HTC-identifiable graph model ideals

Let $G$ be an HTC-identifiable graph, let $\mathcal{M}(G)$ denote its Gaussian structural equation model, and let $J(\mathcal{M}(G))$ be the corresponding model ideal. The generators found in Theorem are generators of $J(\mathcal{M}(G))$. Minimality conjecture. For any HTC-identifiable graph $G$, these generators are minimal. This conjecture extends the established minimality result for graphs whose sets $\{S_v\mid v\in V\}$ have no subset cycles; the general case remains open because the lemma used to prove minimality does not hold for all HTC-identifiable graphs.