Two-global-statement realizability conjecture for gaussoids

Let $N$ be a finite ground set, and let $(I,J|K)$ denote a global conditional-independence statement for pairwise disjoint subsets $I,J,K\subseteq N$. A gaussoid is a conditional-independence structure satisfying the gaussoid axioms. A minimal gaussoid extension of a set of conditional-independence statements is a gaussoid extension minimal under inclusion.

Two-global-statement realizability conjecture. All minimal gaussoid extensions of at most two global conditional-independence statements are realizable.

The claim concerns global conditional-independence statements, which can differ substantially from elementary or local statements when the number of antecedents is bounded. The surrounding discussion notes that the analogous result for local statements does not extend to three antecedents and that the proof does not cover every global configuration; the status of this global two-statement claim is not specified in the source.