The bounded-treewidth conjecture for even-hole-free graphs excluding a 2-forest

Let $t\geq 4$ be an integer, let $H$ be a $2$-forest, and let an $(\mathcal{F})$-free graph mean a graph with no induced subgraph isomorphic to a member of $\mathcal{F}$. The bounded-treewidth conjecture for even-hole-free graphs excluding a 2-forest. Every $($even-hole$,H,K_t)$-free graph has bounded treewidth. This is the if direction of the 2-forest conjecture and is equivalent to the paper's main conjecture; it remains open in general, although the paper proves it for several classes of $H$, including crystals.