Ouyang–Ossona de Mendez–Wood connected tree-depth conjecture

For a graph $H$, let $\operatorname{\overline{td}}(H)$ be its connected tree-depth, and let $\mathcal{M}_H$ be the class of graphs containing no $H$ minor. Let $\chi_{\Delta}(\mathcal{M}_H)$ denote the defective chromatic number. Connected tree-depth conjecture. For every graph $H$,

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

The equality is proved in the source for complete bipartite $H$ and for graphs of connected tree-depth three, but remains unresolved in general.