Facet-defining conjecture for clique inequalities in the two-level graph partitioning polytope

Let $G=(V,E)$ be the graph and let $k,k'$ be the partition parameters, with $r$, $r'$, and $R$ as defined for the corresponding clique inequalities. For a clique $C$, write $y(C)$ and $z(C)$ for the associated clique expressions. Facet-defining conjecture. The following results hold for the convex hull of 2L-PP solutions: the $y$-clique inequalities define facets if and only if $r \ne 0$; the $z$-clique inequalities define facets if and only if $1<R<kk'-1$; and the $(y,z)$-clique inequalities define facets if and only if $r'\ne 0$. This conjecture would characterize exactly when the three newly studied families of clique inequalities are facet-defining, strengthening the preceding necessary conditions for non-domination; the source reports only computational evidence from PORTA, so the general assertions remain open.