The packing-coloring conjecture for bounded-degree graphs

Let $k\geq 3$, and let a $k$-degree graph mean a graph of maximum degree at most $k$. A coloring is $(1^{k-1},2^{k-1},3)$-packing colorable if its vertices can be partitioned into $k-1$ color classes in which distinct vertices have distance at least $2$, $k-1$ color classes in which distinct vertices have distance at least $3$, and one color class in which distinct vertices have distance at least $4$. Packing-coloring conjecture. Every $k$-degree graph is $(1^{k-1},2^{k-1},3)$-packing colorable.

This conjecture refines the known packing-coloring result for subcubic graphs, which are $(1,1,2,2,3)$-packing colorable. The proposed statement concerns all graphs of maximum degree at least three; its resolution is not specified in the source.