Lehel's cycle partition conjecture

Let $G$ be a graph, and let $\overline G$ denote its complement. Lehel's conjecture. The vertices of $G$ can be covered by a cycle $C_1$ in $G$ and a vertex-disjoint cycle $C_2$ in $\overline G$. This is a two-colour cycle-partition problem for complete graphs and is presented as a motivation for studying sparse complements of cycles; the supplied text gives no resolution status.