Hajebi's treewidth–clique boundedness conjecture

Let $\mathcal{G}$ be a hereditary graph class. A graph is $2$-degenerate if every induced subgraph has a vertex of degree at most $2$, and $\mathcal{G}$ is $(\operatorname{tw},\omega)$-bounded if there is a function of the clique number that bounds treewidth throughout $\mathcal{G}$. Hajebi's conjecture. The class $\mathcal{G}$ is $(\operatorname{tw},\omega)$-bounded if and only if $\mathcal{G}$ excludes a complete bipartite graph and every $2$-degenerate graph in $\mathcal{G}$ has bounded treewidth. The conjecture was disproved by Chudnovsky and Trotignon; the source notes that the corresponding pathwidth result is obtained as a consequence of its main theorem.