The rank-+1 hypergraph conjecture for quantum entropy inequalities

Let an $n$-party inequality of the form defined in the source be given, and let an $m$-graph be a hypergraph whose edges have size at most $m$. The inequality contracts on a class of hypergraphs if its contraction map sends every hypergraph in that class to a hypergraph satisfying the inequality.

Rank-$n+1$ hypergraph conjecture. If an $n$-party inequality contracts on $(n+1)$-graphs, then it is a valid inequality on all hypergraphs of finite rank.

This conjecture formalizes the expectation that, after including the purifier, no entanglement structure involving more than $n+1$ indivisible subsystems contributes new information to an $n$-party entropy inequality. Its status is not determined by the supplied text.