Verstraëte's even-cycle subgraph conjecture

Let $k$ and $ell$ be integers with $2\leq \ell<k$. A graph is $C_{2k}$-free if it contains no cycle of length $2k$, and a subgraph is $C_{2\ell}$-free if it contains no cycle of length $2\ell$.

Verstraëte's conjecture. There exists a positive constant $c$ such that every $C_{2k}$-free bipartite graph $G$ has a $C_{2\ell}$-free subgraph $F$ satisfying

$$|E(F)|\geq c|E(G)|.$$

Kühn and Osthus attributed this conjecture to Verstraëte. It would imply that the extremal numbers for forbidding $C_{2k}$ and all cycles of lengths at most $2k$ agree up to a constant factor. The conjecture is refuted: the paper's first result gives arbitrarily large counterexamples for $C_8$ and $C_{10}$, corresponding to $(\ell,k)=(2,4)$ and $(2,5)$.