Class 1 conjecture for $K_5$-minor free graphs of maximum degree six

Let $G$ be a $K_5$-minor free graph with maximum degree $\Delta=6$.

Edge-coloring conjecture. Then $G$ is class $1$.

This conjecture extends the paper's result that every $K_5$-minor free graph with maximum degree at least $7$ is class $1$, and asks whether the analogous conclusion holds at maximum degree six.