Ossona de Mendez–Oum–Wood defective colouring conjecture

For a graph class $\mathcal{G}$, let $\chi_{\Delta}(\mathcal{G})$ be the minimum integer $k$ such that, for some integer $d$, every graph in $\mathcal{G}$ is $k$-colourable with defect $d$. For a graph $H$, let $\mathcal{M}_H$ be the class of $H$-minor-free graphs, and let $\operatorname{\overline{td}}(H)$ denote the connected tree-depth of $H$. Ossona de Mendez–Oum–Wood's conjecture. For every graph $H$,

$$\chi_{\Delta}(\mathcal{M}_H)=\operatorname{\overline{td}}(H)-1.$$

The conjecture is a defective-colouring analogue of Hadwiger's conjecture. The lower bound is proved, and the upper bound is known when $\operatorname{\overline{td}}(H)\leqslant 3$, when $H$ is complete bipartite, and when $H=K_t$; the general upper bound remains open.