The 2-forest conjecture for even-hole-free graphs

For an integer $k\geq 1$, a $k$-forest is a $K_{k+2}$-free chordal graph; in particular, a $2$-forest is a $K_4$-free chordal graph. An even-hole-free graph has no induced cycle of even length at least four, and an $(\mathcal{F})$-free graph contains no induced subgraph isomorphic to a graph in $\mathcal{F}$. The 2-forest conjecture. Given an integer $t\geq 4$ and a $2$-forest $H$, every $($even-hole$,K_t)$-free graph of sufficiently large treewidth contains $H$ if and only if $H$ is a $2$-forest. The only-if direction follows from layered wheels; the substantive if direction is the main open question and is equivalent to the paper's main conjecture.