The Petersen Coloring Conjecture in terms of normal chromatic index

Let $G$ be a bridgeless cubic graph. A normal edge coloring of $G$ is a proper edge coloring in which every edge is poor or rich, and let $\chi_{N}^{\prime}(G)$ be the smallest number of colors in such a coloring.

Petersen Coloring Conjecture.

$$\chi_{N}^{\prime}(G)\leq 5.$$

This is a restatement of the Petersen Coloring Conjecture for bridgeless cubic graphs. The supplied text does not state whether the conjecture has been resolved.