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It was recently found that the stabilizer and hypergraph entropy conescoincide for four parties, leading to a conjecture of their equivalence athigher party numbers. In this note, we show this conjecture to be false byproving new inequalities obeyed by all hypergraph entropy vectors that excludeparticular stabilizer states on six qubits. By further leveraging thisconnection, we improve the characterization of stabilizer entropies and showthat all linear rank inequalities at five parties, except for classicalmonotonicity, form facets of the stabilizer cone. Additionally, by studyingminimum cuts on hypergraphs, we prove some structural properties of hypergraphrepresentations of entanglement and generalize the notion of entanglement wedgenesting in holography.

A Gap Between the Hypergraph and Stabilizer Entropy Cones