Adjacency conjecture for partition m-hemimetrics

Let $V_n$ be the ground set, and let $\alpha(S_1,\dots,S_{m+1})$ and $\alpha(T_1,\dots,T_{m+1})$ be two partition $m$-hemimetrics on $V_n$. Two rays are adjacent when they generate a two-dimensional face of the cone, that is, when they are adjacent in the skeleton of $P_n^m$. Adjacency conjecture for partition m-hemimetrics. The two partition $m$-hemimetrics $\alpha(S_1,\dots,S_{m+1})$ and $\alpha(T_1,\dots,T_{m+1})$ are nonadjacent in the skeleton of $P_n^m$ if and only if there exist six different subsets $S_i,S_j,S_k$ and $T_{i'},T_{j'},T_{k'}$ such that

$$S_i\mathbin{\cup}S_j=T_{k'}$$

and

$$S_k=T_{i'}\mathbin{\cup}T_{j'}.$$

The conjecture holds for $m=1$, when all cut semimetrics are adjacent; for $n-m=2$, when the graph is $J(m+2,2)$; and for $(m,n)=(2,5)$ and $(3,6)$.