NSSW excluded-configuration conjecture for minor-closed classes

Let $\mathcal{X}_{k,c}$ be the graph family defined in the source, and let $\mathcal{M}$ be a minor-closed class. NSSW excluded-configuration conjecture. Every minor-closed class that excludes every graph in $\mathcal{X}_{k,c}$ for some integer $c$ is $k$-colourable with bounded clustering.

Equivalently, for every minor-closed class $\mathcal{M}$, $\chi_{\star}(\mathcal{M})$ should equal the minimum integer $k$ such that $\mathcal{M}\cap\mathcal{X}_{k,c}=\varnothing$ for some integer $c$. The source provides the construction and a rainbow-clique obstruction, but does not establish the conjecture in general.