Facet classification conjecture for the oriented-multicut cone

Let $OMCUT_n$ be the cone of oriented multicuts. A zero-extension of an inequality on $OMCUT_{n-1}$ is the inequality obtained by adding a zero coordinate for the new element. Let $H(b)$ denote a hypermetric inequality, and let $A_n(c_1,\dots,c_{n-2};a,b)$ and $B_n(c_1,\dots,c_{n-2};a,b)$ be the inequalities defined in the paper.

Oriented-multicut facet conjecture. The following inequalities define facets of $OMCUT_n$: (i) zero-extensions of every facet of $OMCUT_{n-1}$, including oriented triangle and non-negativity inequalities; (ii) every hypermetric inequality $H(b)$ except non-oriented triangle inequalities; and (iii) the inequalities $A_n(c_1,\dots,c_{n-2};a,b)$ and $B_n(c_1,\dots,c_{n-2};a,b)$.

The conjecture was checked by computer for $n\leq 7$. Its completeness as a facet description in general remains open.