The positive hypergraph stable-involution conjecture

Let $H$ be an $r$-uniform hypergraph, or $r$-graph. Call $H$ positive when its corresponding hypergraph homomorphism-density functional is nonnegative for every bounded measurable symmetric $r$-variate function. An involution of $V(H)$ is a stable involution if it preserves the edges and, for the partition $V(H)=V_0\sqcup V_+\sqcup V_-$ induced by its fixed points and exchanged pairs, no edge is contained entirely in $V_0$ and no edge intersects both $V_+$ and $V_-$. The positive hypergraph stable-involution conjecture. An $r$-graph is positive if and only if it has a stable involution. The existence of a stable involution is shown to imply positivity, while the converse is presented as a conjectural extension of the positive graph phenomenon to uniform hypergraphs. The source does not provide resolution evidence for this statement, so its status is open.