Relate quadratic stable-set ideals to perfect contractibility

Prove that for every perfect graph G, the following conditions are equivalent: G is perfectly contractile; G contains no odd holes, no antiholes, and no odd prisms as induced subgraphs; and the stable-set toric ideal I_{\mathcal{S}_G} satisfies $\omega(I_{\mathcal{S}_G})=2$.

Background

The conjecture links a structural graph property, a forbidden-subgraph characterization, and an algebraic property of the toric ideal of the stable-set polytope. It is stated for all perfect graphs and is attributed to the third and fourth authors together with Shibata.

The paper proves the claimed equivalence for line perfect graphs after passing to their line graphs, but the general perfect-graph statement remains unresolved.