  It was recently found that the stabilizer and hypergraph entropy cones
coincide for four parties, leading to a conjecture of their equivalence at
higher party numbers. In this note, we show this conjecture to be false by
proving new inequalities obeyed by all hypergraph entropy vectors that exclude
particular stabilizer states on six qubits. By further leveraging this
connection, we improve the characterization of stabilizer entropies and show
that all linear rank inequalities at five parties, except for classical
monotonicity, form facets of the stabilizer cone. Additionally, by studying
minimum cuts on hypergraphs, we prove some structural properties of hypergraph
representations of entanglement and generalize the notion of entanglement wedge
nesting in holography.
