Sárközy's cycle partition conjecture for edge-colored graphs

Let $G$ be a $t$-colored graph, and let $\alpha(G)$ denote its independence number. Let the cycle partition number be the minimum number of vertex-disjoint monochromatic cycles covering $V(G)$. Sárközy's conjecture. The cycle partition number of any $t$-colored graph $G$ is

$$t\alpha(G).$$

The statement is known for $t=1$ and is best possible for $t=2$, but Pokrovskiy's example shows that it is false for every $t\geq3$.