The maximum-average-degree conjecture for 2-distance coloring

Let $G$ be a graph with maximum degree $\Delta(G)=4$ and maximum average degree $\operatorname{mad}(G)<4$. The maximum-average-degree conjecture. asserts

$$\chi^2(G)\leq 13.$$

Here $\chi^2(G)$ is the 2-distance chromatic number. The bound would improve the currently noted upper bound of $15$; the source gives no resolution.