Extreme rays and diameter conjecture for the oriented-multicut skeleton

Let $QMET_n$ be the cone of quasi-semimetrics and $OMCUT_n$ the cone generated by oriented multicuts. Let $G_{OMCUT_n}$ be the skeleton graph of $OMCUT_n$, whose nodes are its extreme rays and whose edges join adjacent extreme rays. A dominating clique is a complete subgraph such that every node is adjacent to at least one of its elements.

Oriented-multicut skeleton conjecture. (i) All oriented multicuts are extreme rays of $QMET_n$. (ii) The oriented cuts form a dominating clique in $G_{OMCUT_n}$; consequently, the diameter of $G_{OMCUT_n}$ is $2$ or $3$ for every $n\geq 4$.

The assertions were presented as conjectural general results about the oriented-multicut cone; the surrounding text does not provide a general proof or resolution.