Gastineau–Togni conjecture on packing colorings of subcubic graphs

A packing $(1,1,2,3)$-coloring is a partition of the vertex set into four classes whose pairwise vertex distances are respectively at least $2$, $2$, $3$, and $4$. Gastineau–Togni conjecture. Every connected subcubic graph except the Petersen graph is packing $(1,1,2,3)$-colorable. The paper says that this was posed as a conjecture because the authors believe it is true; no resolution is supplied in the candidate context.