The H-union-K_2 perfect-divisibility conjecture

Let $H$ be a graph, let $K_2$ be the complete graph on two vertices, and let $k\in\mathbb{N}_{>2}$. Assume that every $H$-free graph is perfectly $(k-2)$-divisible. The H-union-K_2 conjecture. Then every $(H\cup K_2)$-free graph is perfectly $k$-divisible. This conjecture proposes a general mechanism for lifting perfect divisibility from $H$-free graphs to graphs forbidding the disjoint union $H\cup K_2$; the supplied source gives no resolution status.