The near-bipartite characterization conjecture for ${\operatorname{LS}}_+$-perfect graphs

For a graph $G$, let $\operatorname{STAB}(G)$ be its stable set polytope, $\operatorname{NB}(G)$ the near-bipartite polyhedral relaxation, and ${\operatorname{LS}}_+(G)$ the Lovász–Schrijver positive semidefinite relaxation. A polyhedron is polyhedral if it can be described by finitely many linear inequalities. The near-bipartite characterization conjecture. For every graph $G$, the following four statements are equivalent: (1) $\operatorname{STAB}(G)=\operatorname{NB}(G)$; (2) ${\operatorname{LS}}_+(G)=\operatorname{STAB}(G)$; (3) ${\operatorname{LS}}_+(G)=\operatorname{NB}(G)$; and (4) ${\operatorname{LS}}_+(G)$ is polyhedral. This would characterize the graphs on which the Lovász–Schrijver operator reaches the stable set polytope using the near-bipartite relaxation; the paper presents it as an open conjecture.