Induced-ridge-graph conjecture for m-hemimetric cones

Let $HM_n^m$, $NHM_n^m$, and $P_n^m$ be the corresponding cones of $m$-hemimetrics, and let the ridge graph of a cone have the facets as vertices, with adjacency when two facets meet in a ridge. Induced-ridge-graph conjecture. The ridge graph of $HM_n^m$ is an induced subgraph of the ridge graph of $NHM_n^m$, and the ridge graph of $NHM_n^m$ is an induced subgraph of the ridge graph of $P_n^m$. In $NHM_n^m$, the two vertex orbits are $F_1$ and $F_2$, consisting respectively of $(n-m-1)\binom{n}{m+1}$ simplex inequalities and $\binom{n}{m+1}$ nonnegativity inequalities. The conjecture concerns the general inclusion pattern of these ridge graphs; the source gives computational observations for small parameters but no general proof.