Even cycles in graphs avoiding longer even cycles

A conjecture of Verstraëte states that for any fixed $\mathrm{\ell }<k$ there exists a positive constant $c$ such that any $C_{2k}$-free graph $G$ contains a $C_{2\mathrm{\ell }}$-free subgraph with at least $c\mathrm{\mid }E(G)\mathrm{\mid }$ edges. For $\mathrm{\ell }=2$, this conjecture was verified by Kühn and Osthus. We show that $C_6$ and $C_{2k}$ satisfy the conjecture for all odd $k$, but observe that a recent construction of a dense $C_{10}$-free subgraph of the hypercube yields a counterexample to the conjecture for $C_8$ and $C_{10}$.