Sullivant's saturation conjecture for Gaussian DAG models

Let $G$ be a directed acyclic graph, let $P_G$ be the vanishing ideal of its Gaussian model, and let $J_G$ be its global conditional independence ideal. Define

$$S_V = \Set{ \prod_{A \subseteq V} |\Sigma_A|^{k_A}: k_A\in \mathbb{N}}.$$

Sullivant's saturation conjecture. The vanishing ideal satisfies

$$P_G = J_G: S_V.$$

This conjecture concerns the relationship between the parametric definition of a Gaussian DAG model and the polynomial constraints arising from its global Markov properties. The source states that the paper proves the result, but the supplied status is unknown; its resolution should be checked.