Adjacency and diameter conjecture for the ridge graph of the quasi-semimetric cone

Let $QMET_n$ be the cone of quasi-semimetrics, with triangle facets, non-negativity facets $NN_{ij}$, and ridge graph $G^{*}_{QMET_n}$ whose nodes are the facets of $QMET_n$ and whose edges join adjacent facets. Two vectors are conflicting if they have components with nonzero values of different signs.

Adjacency and diameter conjecture. (i) A triangle facet is adjacent to a facet if and only if they are non-conflicting. (ii) The non-negativity facets $NN_{ij}$ and $NN_{i'j'}$ are adjacent if and only if neither $i'=j$ nor $j'=i$. (iii) The diameter of $G^{*}_{QMET_n}$ is $2$. (iv) The ridge graph $G^{*}_{QMET_n}$ is an induced subgraph of $G^{*}_{OMCUT_n}$.

These assertions were checked by computer for $n\leq 7$. The diameter assertion follows formally from the proposed adjacency criteria together with the preceding non-adjacency results, but the general adjacency and induced-subgraph claims remain open.