Generic identifiability conjecture for all non-complete graph families

Let $\mathcal{G}$ be any family of non-complete graphs, and let $M_G$ denote the homoscedastic linear structural equation model associated with $G\in\mathcal{G}$. Generic identifiability conjecture. The graph parameter $G$ of the family $\{M_G\}_{G\in\mathcal{G}}$ is generically identifiable. The claim would extend the paper's established identifiability results from particular graph families to every family of non-complete graphs. It is stated without a proof or resolution.