Claw-free cubic graphs are (1, 1, 1, 3)-packing edge-colorable

For a non-decreasing positive integer sequence $S=(s_1,\dots ,s_k)$, an $S$-packing edge-coloring of a graph $G$ is a partition of the edge set of $G$ into subsets $E_1,\dots ,E_k$ such that for each $1\le i\le k$, the distance between any two distinct edges $e_1,e_2\in E_i$ is at least $s_i+1$. Gastineau and Togni conjectured that cubic graphs, except the Petersen and Tietze graphs, admit $(1,1,1,3)$-packing edge-colorings. In this paper, we prove that every claw-free cubic graph admits such a coloring.