NSSW19's clustered colouring conjecture for minor-free graphs

For a graph $H$, let $\mathcal{M}_H$ be the class of graphs with no $H$ minor. Write $\raisebox{1.55pt}{\scalebox{1.25}{\chi}}_{\star}(\mathcal{M}_H)$ for its clustered chromatic number and $\operatorname{\overline{td}}(H)$ for the connected treedepth of $H$. NSSW19's conjecture. For every graph $H$,

$$\raisebox{1.55pt}{\scalebox{1.25}{\chi}}_{\star}(\mathcal{M}_H)\leqslant 2\operatorname{\overline{td}}(H)-2.$$

The conjecture is an upper-bound analogue of the defective-colouring conjecture above. The source states that it remains open, while the paper proves matching lower-bound examples for suitable graphs of each connected treedepth.