Block structure conjecture for graphs avoiding consecutive even cycle lengths

Let $t\geq1$ and let $G$ be a graph with a maximum number of edges among graphs that do not contain cycles of $t$ consecutive even lengths. A block is a maximal connected subgraph with no cut vertex. Block structure conjecture. Every block of $G$ is a complete graph of order at most $2t+1$. The conjecture is stated as a strengthening of Thomassen's conjecture and is known in the source only for $t=1$; its general case remains open.