The even-hole-free graph and K4-free chordal graph conjecture

Let $t\geq 4$ be an integer, let $H$ be a graph, and call $H$ $K_4$-free chordal if it is chordal and contains no induced subgraph isomorphic to $K_4$. An even-hole-free graph has no induced cycle of even length at least four, and $\operatorname{tw}(G)$ denotes the treewidth of $G$. The even-hole-free graph and K4-free chordal graph conjecture. For every $t\geq 4$ and every $K_4$-free chordal graph $H$, every even-hole-free graph of sufficiently large treewidth has an induced subgraph isomorphic to $K_t$ or $H$. This is the paper's main conjecture; it is known for $t=4$, for coned forests, and for crystals, but remains open in general.