Packing chromatic number conjecture for 0-saturated subcubic graphs

Let $G$ be a $0$-saturated subcubic graph, and let $g_3(G)$ denote its local girth parameter. Packing chromatic number conjecture. If

$$g_3(G)\leq 4,$$

then

$$\chi_{\rho}(G)\leq 4.$$

The paper proves the corresponding upper bound $\chi_{\rho}(G)\leq 6$ and reports no example with packing chromatic number at least $5$; the claimed improvement to $4$ remains open.