Gaussoid axioms suffice for CI implication in singly constrained Gaussian DAG models

Let $\mathcal{G}$ be a directed acyclic graph (DAG), and let $i \perp\!\!\!\perp j \mid K$ be a conditional independence statement such that $i \not\perp_{\mathcal{G}} j \mid K$. Denote by $\mathcal{M}_{\mathcal{G}, i \perp\!\!\!\perp j\mid K}$ the Gaussian model associated with $\mathcal{G}$ together with this additional conditional independence statement. Let $\operatorname{global}(\mathcal{G})$ be the set of global Markov conditional independence statements of $\mathcal{G}$.

Gaussoid-axioms sufficiency conjecture. If $a \perp\!\!\!\perp b \mid C$ holds for every covariance matrix $\Sigma \in \mathcal{M}_{\mathcal{G}, i \perp\!\!\!\perp j\mid K}$, then $a \perp\!\!\!\perp b \mid C$ is implied by applying the gaussoid axioms to

$$\operatorname{global}(\mathcal{G}) \cup \{i \perp\!\!\!\perp j \mid K\}.$$

The conjecture is verified in the paper for models involving four random variables, but its validity for arbitrary numbers of variables remains open.