Cocks's hereditary treewidth–clique boundedness conjecture

Let $\mathcal{G}$ be a hereditary graph class. A graph $H$ is $H$-free in $\mathcal{G}$ when graphs in the relevant subclass have no induced subgraph isomorphic to $H$. The class $\mathcal{G}$ is $(\operatorname{tw},\omega)$-bounded if there is a function of clique number that bounds treewidth throughout $\mathcal{G}$. Cocks's conjecture. The class $\mathcal{G}$ is $(\operatorname{tw},\omega)$-bounded if and only if $\mathcal{G}$ excludes a complete bipartite graph and, for every non-complete graph $H\in\mathcal{G}$, the class of all $H$-free graphs in $\mathcal{G}$ is $(\operatorname{tw},\omega)$-bounded. The source identifies this conjecture as still open and says that its pathwidth analogue follows from the paper's main theorem.