The P5-free perfect divisibility conjecture

Let $P_5$ denote the chordless path on five vertices. 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 $(G[B])<(G)$.

P5-free conjecture. Every $P_5$-free graph is perfectly divisible.

The conjecture is proposed in connection with the paper's clique-cutset results. Chudnovsky and Sivaraman proved the related result that $(P_5,\operatorname{bull})$-free graphs are perfectly divisible; the stated conjecture remains open.