The sparse induced-subgraph conjecture for even-hole-free graphs

Let $\operatorname{tw}(G)$ denote the treewidth of $G$. The sparse induced-subgraph conjecture for even-hole-free graphs. For every integer $t\geq 1$, every even-hole-free graph of sufficiently large treewidth has an induced subgraph of treewidth $t$ which is either complete or $K_4$-free. The conjecture would imply the main conjecture by combining it with the paper's theorem for even-hole-free, $K_4$-free graphs; it remains open as stated.