Verstraëte's conjecture on consecutive even cycles

Let $G$ be an $n$-vertex graph, and let $e(G)$ denote its number of edges. Say that $G$ contains $k$ cycles of consecutive even lengths when it contains such a collection of cycles. Verstraëte's conjecture. If $G$ is an $n$-vertex graph not containing $k$ cycles of consecutive even lengths, then

$$e(G) \leq \frac{1}{2}(2k+1)(n-1),$$

with equality if and only if every block of $G$ is a clique of order $2k+1$. The conjecture generalizes extremal results on the existence of consecutive even cycle lengths; its cases $k=1$ and $k=2$ are known, while the general statement remains open.