The 2-tone edge coloring conjecture for cubic graphs excluding $K_4$

Let $G$ be a cubic graph that does not contain $K_4$, and let $\tau'_2(G)$ denote its 2-tone edge chromatic number, namely the minimum number of colors needed for a 2-tone edge coloring of $G$. The strengthened 2-tone edge coloring conjecture.

$$\tau'_2(G)\le 8.$$

This strengthens the preceding conjecture by excluding $K_4$, which requires nine colors. The authors propose it because they believe that $K_4$ is the unique graph requiring nine colors.