Sudakov–Verstraëte's extremal block conjecture for consecutive even cycles

Let $G$ be a graph with the maximum possible number of edges among graphs that do not contain $k$ cycles of consecutive even lengths. A block is a maximal connected subgraph with no cut vertex. Sudakov–Verstraëte's conjecture. Every block of $G$ is a clique of order at most $2k+1$. This is presented as a strengthening related to Thomassen's conjecture; the source proves the case $k=2$, while the general statement remains open.