Exceptional packing coloring conjecture for claw-free subcubic graphs

Let $G$ be a claw-free subcubic graph, and let $\mathcal{G}_{11}$ be the graph specified in the paper. Exceptional packing coloring conjecture. If $G\neq\mathcal{G}_{11}$, then $G$ is $(1,1,3,3,3)$-packing colorable. The paper exhibits $\mathcal{G}_{11}$ as a counterexample and conjectures that it is the unique exception; this classification remains open.