Conjecture on the 2-tone chromatic number of cubic Halin graphs

Let $G$ be a cubic Halin graph of order $n$, where a Halin graph is formed from a tree with no vertices of degree two and a cycle joining its leaves in their planar cyclic order. The conjecture. Every cubic Halin graph of order $n\ge 6$ is $2$-tone $6$-colorable.

This would improve the established upper bound for cubic Halin graphs and, since cubic Halin graphs are $K_4-e$-free, would imply the third part of the Bickle–Phillips conjecture for this class. No cubic Halin graph requiring seven colors is known in the supplied text.