Conjectured ridge graphs of partial metric and weighted quasi-metric cones

Let $\text{\sl PMET}_n$ and $\text{\sl WQMET}_n$ be the partial-metric and weighted quasi-metric cones from the source, and let ${{\operatorname{Ri}}}$ denote their ridge graphs. The vertices $L_{ii}$, $M_{ij}$, and $Tr_{ij,k}$ are the corresponding facet classes; two $Tr$ facets conflict when they have values of different signs on a common position among their indexed coordinates.

Ridge-graph conjecture. (i) ${{\operatorname{Ri}}}(\text{\sl PMET}_n)$ has diameter $2$, with all non-adjacencies given by $L_{ii}\nsim M_{ik}$; $M_{ij}\nsim M_{ji},M_{ki},M_{jk},Tr_{ij,k}$; and $Tr_{ij,k}\nsim Tr_{i'j',k'}$ when they conflict. (ii) ${{\operatorname{Ri}}}(\text{\sl WQMET}_n)$ has diameter $2$ and is obtained from ${{\operatorname{Ri}}}(\text{\sl PMET}_n)$ by deleting the vertices $L_{ii}$.

These claims describe the full non-adjacency patterns in the ridge graphs of two families of polyhedral cones. The source calls the proofs tedious but easy and supplies no evidence that the conjectures have been resolved.