Asymptotic Sárközy cycle-cover conjecture for edge-colored graphs

Let $G$ be a $t$-colored graph with independence number $\alpha(G)=\alpha$, and let $n=|V(G)|$. Asymptotic Sárközy cycle-cover conjecture. There exists a constant $c=c(\alpha,t)$ such that $t\alpha$ vertex-disjoint monochromatic cycles of $G$ cover at least

$$n-c$$

vertices. The conjecture is open in the source, including for $t=2$; Pokrovskiy's example implies that necessarily $c\geq\alpha$.