The linear-forest conjecture for intersectionwise self-$\chi$-guarding graph classes

A linear forest is a forest in which every component is a path. For a graph $H$, an $H$-free graph is a graph with no induced subgraph isomorphic to $H$; a class is intersectionwise self-$\chi$-guarding when it has the property defined in the paper.

Linear-forest conjecture. If $H$ is a linear forest, then the class of $H$-free graphs is intersectionwise self-$\chi$-guarding.

The preceding results establish this property for stars and for $P_{4}$-free graphs, while the paper's corollary shows that any graph $H$ whose $H$-free class is intersectionwise self-$\chi$-guarding must be a linear forest or a star. The conjecture addresses the remaining linear-forest cases.