The even-hole-free perfect divisibility conjecture

A hole is a chordless cycle with at least four vertices, and an even hole is a hole of even length. A graph is perfectly divisible if every induced subgraph with at least one edge has a partition $(A,B)$ such that the induced graph on $A$ is perfect and $\omega(G[B])<\omega(G)$.

Even-hole-free conjecture. Every even-hole-free graph is perfectly divisible.

The source notes that every even-hole-free graph contains a bisimplicial vertex, where a bisimplicial vertex has a neighbourhood covered by two cliques. The perfect divisibility conjecture remains open.