The full-support facet conjecture for ${\operatorname{LS}}_+$-perfect graphs

Let ${\operatorname{LS}}_+(G)$ denote the Lovász–Schrijver positive semidefinite relaxation of the stable set polytope. A graph is ${\operatorname{LS}}_+$-perfect when ${\operatorname{LS}}_+(G)=\operatorname{STAB}(G)$, and a facet is full-support when its defining inequality involves every vertex of $G$. The full-support facet conjecture. If a graph is ${\operatorname{LS}}_+$-perfect and its stable set polytope has a full-support facet-defining inequality, then the graph is near-bipartite. This is presented as an equivalent formulation of the paper's open characterization problem for ${\operatorname{LS}}_+$-perfect graphs.