Facial edge-coloring conjecture for plane graphs

Let $G$ be a plane graph, and let $u$ be a positive integer. An $\nu$-facial edge-coloring of $G$ is an edge-coloring in which all edges on every facial trail of length at most $\nu+1$ receive distinct colors. Facial edge-coloring conjecture. Every plane graph admits an $\nu$-facial edge-coloring with at most

$$3\nu+1$$

colors for every $\nu\geq 1$. This conjecture is known for $\nu\leq 3$: the cases $\nu\leq 2$ were previously confirmed, and the paper proves the case $\nu=3$; the general case remains open.