Odd-minor clustered and defective colouring conjecture

Let $H$ be a graph, let $\mathcal{G}_H^{\text{odd}}$ be the class of graphs with no $H$ odd-minor, let $\operatorname{\overline{td}}(H)$ denote the connected tree-depth of $H$, let $\chi_{\star}(\mathcal{G}_H^{\text{odd}})$ be the clustered chromatic number, and let $\chi_{\Delta}(\mathcal{G}_H^{\text{odd}})$ be the defective chromatic number. Odd-minor clustered and defective colouring conjecture. For every graph $H$,

$$\chi_{\star}(\mathcal{G}_H^{\text{odd}})\leqslant 2\operatorname{\overline{td}}(H)-2$$

and

$$\chi_{\Delta}(\mathcal{G}_H^{\text{odd}})=\operatorname{\overline{td}}(H)-1.$$

The paper proves the weaker upper bound $\chi_{\star}(\mathcal{G}_H^{\text{odd}})\leqslant 3\cdot 2^{\operatorname{\overline{td}}(H)}-4$ and places the conjecture as the proposed improvement; its status is not resolved in the source.