Dvořák et al.'s six-color conjecture for subcubic graphs

Let $G$ be a graph of maximum degree at most three, and let $\chi^\prime_s(G)$ denote its star chromatic index, the smallest number of colors in a proper edge coloring in which no path or cycle with four edges is bicolored. Dvořák et al.'s conjecture.

$$\chi^\prime_s(G)\leq 6.$$

This conjecture seeks a universal six-color bound for star edge coloring of subcubic graphs. The supplied text reports the conjecture but gives no resolution, so its status is open.