Conjectured skeletons and non-adjacencies for partial metric and weighted quasi-metric cones

Let $S(n,2)=2^{n-1}-1$ be the Stirling number counting partitions of an $n$-element set into two nonempty blocks. For a polyhedral cone, write ${\operatorname{Sk}}$ for its skeleton, $K_m$ for the complete graph on $m$ vertices, $K_{a,b}$ for the complete bipartite graph, and $+$ for disjoint union. Let $\text{\sl OCUT}_n$, $\text{\sl WQMET}_n$, and $0,1\text{-}\text{\sl dWMET}_n$ denote the corresponding cones used in the source; $\delta'(S)$ and $w'$ denote the indicated cut and weight vertices.

Skeleton conjecture. (i) ${{\operatorname{Sk}}}(\text{\sl OCUT}_n)=K_{2S(n,2)}$ and is a subgraph of ${{\operatorname{Sk}}}(\text{\sl WQMET}_n)$. (ii) The complement of ${{\operatorname{Sk}}}(0,1\text{-}\text{\sl dWMET}_n)$ is $K_{1,S(n,2)}+S(n,2)K_2$; its skeleton has diameter $2$, and all non-adjacencies are of the forms $(((0));(1))\nsim(\delta'(S);(0))$ and $(\delta'(S);w')\nsim(\delta'(S);w')$.

The claims concern the adjacency structure of skeleton graphs of cones of oriented cuts, weighted quasi-metrics, and discrete weighted metrics. The paper says that the proofs should be tedious but easy, but the supplied material gives no resolution evidence.