The normal 5-edge-coloring conjecture for bridgeless cubic graphs

Let $G$ be a cubic graph. A normal edge-coloring is a proper edge-coloring in which every edge $uv$ satisfies $|S_c(u)\cup S_c(v)|\in\{3,5\}$, where $S_c(v)$ is the set of colors on the edges incident with $v$. Let $\chi'_N(G)$ denote the minimum number of colors in a normal edge-coloring of $G$. Normal 5-edge-coloring conjecture. For any bridgeless cubic graph $G$, $\chi'_N(G)\leq 5$. By Jaeger's equivalence, this is a reformulation of the Petersen coloring conjecture and remains open in general.