The perfect-graph characterization of quadratic stable-set toric ideals

Let $G$ be a perfect graph. Let $\mathcal{S}_G$ denote its stable-set polytope and $I_{\mathcal{S}_G}$ the associated toric ideal. Use the notions of perfectly contractile graph, odd hole, antihole, and odd prism as defined above.

Stable-set ideal conjecture. The following conditions are equivalent:

1. $G$ is perfectly contractile.
2. $G$ contains no odd holes, no antiholes, and no odd prisms.
3. $\omega(I_{\mathcal{S}_G})=2$.

The conjecture links the structural characterization of perfectly contractile perfect graphs with quadratic generation of the stable-set polytope's toric ideal; the supplied text gives no resolution.