The LS+-perfect graph conjecture

Let $G$ be a graph. Write ${\rm STAB}(G)$ for its stable set polytope and ${\rm ASTAB}^*(G)$ for the polyhedral relaxation associated with joined a-perfect graphs. A graph is ${\rm LS}_+$-perfect when ${\rm LS}_+(G)={\rm STAB}(G)$, where ${\rm LS}_+(G)$ is the Lovász–Schrijver positive semidefinite relaxation applied to ${\rm ESTAB}(G)$.

${\rm LS}_+$-perfect graph conjecture. A graph $G$ is ${\rm LS}_+$-perfect if and only if

$${\rm STAB}(G)={\rm ASTAB}^*(G).$$

Equivalently, the conjecture asserts that ${\rm LS}_+$-perfect graphs coincide with joined a-perfect graphs, with ${\rm ASTAB}^*(G)$ serving as the corresponding polyhedral relaxation of ${\rm STAB}(G)$.