Karthick–Kishore–Sahu conjecture on perfect divisibility of fork-free graphs

A graph is perfectly divisible if, for every induced subgraph $H$, its vertex set can be partitioned into $A$ and $B$ such that $H[A]$ is perfect and $\omega(H[B])<\omega(H)$. A fork is obtained from the claw $K_{1,3}$ by subdividing one edge once. Karthick–Kishore–Sahu conjecture. The class of fork-free graphs is perfectly divisible. Perfect divisibility yields a quadratic upper bound on chromatic number in terms of clique number. The parser supplies no resolution evidence for this conjecture, so its status is left open.