A conjecture of Verstra\"ete on vertex-disjoint cycles

Answering a question of H\"aggkvist and Scott, Verstra\"ete proved that every sufficiently large graph with average degree at least $k^2+19k+10$ contains $k$ vertex-disjoint cycles of consecutive even lengths. He further conjectured that the same holds for every graph $G$ with average degree at least $k^2+3k+2$. In this paper we prove this conjecture for $k\geq 19$ when $G$ is sufficiently large. We also show that for any $\epsilon>0$ and large $k\geq k_\epsilon$, average degree at least $k^2+3k-2+\epsilon$ suffices, which is asymptotically tight for infinitely many graphs.