Ridge-graph structure conjecture for NHM m-hemimetric cones

Let $NHM_n^m$ be the cone of nonnegative $m$-hemimetrics. Denote by $T_{i_1\dots i_{m+1},i_{m+2}}$ an $(m+1)$-simplex facet and by $N_{i_1\dots i_{m+1}}$ a nonnegativity facet. Let $F_1$ and $F_2$ be the two facet orbits, and write $G(F_2)$ for the subgraph induced by $F_2$. Ridge-graph structure conjecture. The ridge graph of $NHM_n^m$ satisfies: (i) $T_{i_1\dots i_{m+1},i_{m+2}}$ is adjacent to every other facet except the following $m+2$ facets: all other $(m+1)$-simplex facets with the same support and $N_{i_1\dots i_{m+1}}$; (ii)

$$G(F_2)=\overline{J(n,3)}\quad\text{for }m=2,$$

and

$$G(F_2)=K_{\binom{n}{m+1}}\quad\text{for }m\geq 3.$$

Consequently, the restriction to $F_1$ is $G(F_1)=K_{m+2,\dots,m+2}$, and the conjecture would imply that the ridge graph of $NHM_n^m$ has diameter $2$. The diameter-$2$ conclusion is known for $m=1$, where $NHM_n^1=MET_n$.