Bounded treedepth for even-cycle-subgraph-free graphs of diameter three

Let $r\geq 4$. A graph is $C_{2r}$-subgraph-free if it contains no subgraph isomorphic to the cycle $C_{2r}$. The diameter of a graph is the maximum distance between two vertices. The even-cycle treedepth conjecture. For every $r\geq 4$, the class of $C_{2r}$-subgraph-free graphs of diameter at most $3$ has bounded treedepth.

The conjecture would complete the classification of bounded treedepth for even-cycle-subgraph-free graphs of diameter three. Boundedness is established for $r=4$ by an involved case analysis and for $r\in\{5,\ldots,12\}$ with computer assistance, while the general case remains open.