The correspondence coloring conjecture for $K_5$-minor-free graphs

Let $G$) be a $K_5$-minor-free graph, and let $\mathbf{M}$ be a correspondence $6$-cover of $G$. A collection of $6$ colorings is disjoint if they are pairwise disjoint as $\mathbf{M}$-colorings.

Correspondence coloring conjecture. The graph $G$ has $6$ disjoint $\mathbf{M}$-colorings. Equivalently,

$$\chi^{\star}_{c}(\mathcal{K}_5)=6,$$

where $\mathcal{K}_5$ is the class of all $K_5$-minor-free graphs and $\chi^{\star}_{c}$ denotes the disjoint correspondence coloring number.

The best known upper bound is $\chi^{\star}_{c}(\mathcal{K}_5)\leqslant 8$, so the conjecture would determine the first open case after $K_4$-minor-free graphs. It is motivated by the known bounds for smaller excluded cliques and by the fact that every $K_5$-minor-free graph has maximum average degree less than $6$.