Coloring the square of a sparse graph $G$ with almost $\mathrm{\Delta }(G)$ colors

For a graph $G$, let $G^2$ be the graph with the same vertex set as $G$ and $xy\in E(G^2)$ when $x\mathrm{\ne }y$ and $d_G(x,y)\le 2$. Bonamy, L'ev^{e}que, and Pinlou conjectured that if $mad(G)<4-\frac{2}{c+1}$ and $\mathrm{\Delta }(G)$ is large, then ${\chi }_{\mathrm{\ell }}(G^2)\le \mathrm{\Delta }(G)+c$. We prove that if $c\ge 3$, $mad(G)<4-\frac{4}{c+1}$, and $\mathrm{\Delta }(G)$ is large, then $\chi_\ell(G^2) \leq \Delta(G) + c$. Dvo\v{r}'ak, Kr'{a}\soft{l}, Nejedl'{y}, and \v{S}krekovski conjectured that $\chi (G^2)\le \mathrm{\Delta }(G)+2$ when $\mathrm{\Delta }(G)$ is large and $G$ is planar with girth at least $5$; our result implies $\chi (G^2) \leq \Delta(G) +6$.