Abstract
A (1, 1, 2, k)-packing coloring of a graph G is a partition of V (G) into two independent sets, a 2-packing, and a k-packing. Recently, the question was posed in [A short proof that every claw-free cubic graph is (1, 1, 2, 2)-packing colorable, arXiv:2512.24001v1] as to whether every claw-free cubic graph is (1, 1, 2, 3)-packing colorable. We provide an answer in the affirmative in the case that G is a diamond-free, claw-free cubic graph.