H-free-complement monochromatic path-cover conjecture

Let $H$ be a graph with chromatic number $k+1$, and let $G$ be an $t$-edge-colored graph on $n$ vertices such that $H$ is not a subgraph of the complement $\overline{G}$. H-free-complement path-cover conjecture. There exists a constant $c=c(H,k,t)$ such that $kt$ vertex-disjoint monochromatic paths of $G$ cover at least

$$n-c$$

vertices. The conjecture extends cycle- and path-partition questions from complete graphs to graphs whose complements exclude a fixed graph; the source proves special cases but leaves the general statement open.