Adjacency and diameter conjecture for the oriented-multicut skeleton

Let $OMCUT_n$ be the cone of oriented multicuts, and let $G_{OMCUT_n}$ be its skeleton graph. For an oriented partition $A$, write $A^T$ for its reversal, and write $A<B$ when $A$ is a proper refinement of $B$; write $A\mathrel{\dot{<}}B$ when every part of $A$ is a proper subset of a part of $B$. Let $p'(n)$ denote the number of oriented-multicut extreme rays.

Oriented-multicut adjacency conjecture. (i) An oriented multicut $\delta'(A)$ is not adjacent to every oriented multicut $\delta'(B)$ such that $B<A^T$. (ii) The orbit represented by the oriented cut $\delta'(\{1\},\{2,\dots,n\})$ is the unique orbit whose extreme rays are not adjacent only to the oriented multicuts described in (i); its total adjacency is $p'(n)-p'(n-1)-1$, and this is maximal. (iii) An extreme ray in the orbit represented by $\delta'(\{1,2\},\{3,\dots,n\})$ is not adjacent only to oriented multicuts $\delta'(B)$ for which either $B<(\{1,2\},\{3,\dots,n\})^T$, or $B$ is a cyclic shift of some $C$ with $C\mathrel{\dot{<}}(\{1,2\},\{3,\dots,n\})$. (iv) The diameter of $G_{OMCUT_n}$ is $2$.

The source gives these as conjectural adjacency properties; no general verification or resolution is supplied in the provided text.