Bickle–Phillips conjecture on the 2-tone chromatic number of cubic graphs

Let $G$) be a cubic graph. The Bickle–Phillips conjecture.

$$\tau_2(G)\le 8.$$

If $G$ does not contain $K_4$, then

$$\tau_2(G)\le 7.$$

If $G$ does not contain $K_4-e$, then

$$\tau_2(G)\le 6.$$

The first assertion is known, while the third is disproved by the Heawood graph, which does not contain $K_4-e$; the status of the second assertion is not specified here.