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 $1 \leq i \leq k$, 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.