Karthick et al.'s perfect divisibility conjecture for fork-free graphs

A graph $G$ is fork-free if it has no induced subgraph isomorphic to the graph obtained from $K_{1,3}$ by subdividing one edge once. A graph is perfectly divisible if, for each induced subgraph $H$ of $G$, there is a partition $V(H)=A\mathbin{\dot\cup}B$ such that $H[A]$ is perfect and $\omega(H[B])<\omega(H)$. Karthick et al.'s conjecture. The class of fork-free graphs is perfectly divisible. Perfect divisibility provides a structural route to controlling the chromatic number of hereditary graph classes. The paper states this conjecture as previously proposed by Karthick et al.; no resolution is given in the supplied text.