Charpentier's coloring conjecture for squares of graphs with maximum average degree below 4

Let $G$ be a graph, let $\Delta(G)$ denote its maximum degree, let $\operatorname{mad}(G)$ denote its maximum average degree, and let $G^2$ be its square, with chromatic number $\chi(G^2)$. Charpentier's conjecture. There exists an integer $D$ such that every graph $G$ with $\Delta(G) \ge D$ and $\operatorname{mad}(G) < 4$ has

$$\chi(G^2) \le 2\Delta(G).$$

The conjecture was disproved by Kim and Park, who showed that for every positive integer $D$ there is a graph satisfying the stated degree and maximum-average-degree conditions with $\chi(G^2)=2\Delta(G)+2$.