Karthick–Kumar–Sivaraman conjecture for fork-free graphs

Let $G$ be a graph. A graph is perfectly divisible if, for every induced subgraph $H$, the vertex set $V(H)$ can be partitioned into sets $A$ and $B$ such that $H[A]$ is perfect and $\omega(H[B])<\omega(H)$. Karthick–Kumar–Sivaraman conjecture. Every fork-free graph is perfectly divisible. Karthick et al. proposed this conjecture after proving perfect divisibility for some subclasses of fork-free graphs; it remains open.