Clique–Stable Set separation conjecture for $H$-free graphs

Let $H$ be a fixed graph, and let an $H$-free graph be a graph with no induced subgraph isomorphic to $H$. A CS-separator is a family of cuts separating every disjoint clique from every stable set. Clique–Stable Set separation conjecture for $H$-free graphs. The Clique–Stable Set separation conjecture is true on $H$-free graphs.

The source presents this as a proposed restriction of the general separation conjecture, motivated by evidence for perfect graphs and other graph classes. No resolution is given in the supplied text.