Conjecture that P₅-free graphs are path-perfect

Let $G$ be a graph. A graph is path-perfect if its chromatic number equals the maximum chromatic number of a subgraph spanned by an odd cycle, that is, $\chi(G)=r(G)$. P₅-free path-perfectness conjecture. If $G$ is a $P_5$-free graph, then $G$ is path-perfect. The source describes this as an even stronger conjecture than path-perfectness for the two preceding forbidden-subgraph classes; it remains open.