Irreducible non-realizability conjecture for spherical occlusion diagrams

Let a Spherical Occlusion Diagram (SOD) be a finite collection of geodesic arcs on the sphere satisfying the axioms in the paper, and call an SOD irreducible if no proper subset is an SOD. For a polyhedron $\mathcal P$ and a viewpoint $v$ that sees no vertices of $\mathcal P$, let $S_\mathcal P$ denote its visibility map.

Irreducible non-realizability conjecture. There is an irreducible SOD satisfying the axioms that is not the visibility map $S_\mathcal P$ of any polyhedron $\mathcal P$ with respect to any viewpoint $v$ that sees no vertices of $\mathcal P$.

The converse of the realizability result for polyhedral visibility maps is known to fail for general SODs, but the cited counterexample is not irreducible. The conjecture asserts that non-realizability persists even under irreducibility; it is refuted by the counterexample mentioned in the parser's status evidence.