Five-color packing coloring conjecture for 2-saturated subcubic graphs

Let $G$ be a $2$-saturated subcubic graph. Five-color packing coloring conjecture. The graph $G$ is $(1,2,2,2,2)$-packing colorable. The paper proves a four-color variant when $g_3(G)=3$ and reports no counterexample to this assertion for the next local-girth case; the general assertion remains open.