Cranston's maximum-average-degree conjecture for odd colorings

Let $G$ be a finite simple graph, let $\operatorname{mad}(G)$ be the maximum of $\frac{2|E(H)|}{|V(H)|}$ over all non-empty subgraphs $H$ of $G$, and let $\chi_o(G)$ be the least positive integer $c$ for which $G$ has a proper $c$-coloring in which every vertex of positive degree has some color appearing an odd number of times among its neighbors. Cranston's conjecture. For $c\geq 4$, if

$$\operatorname{mad}(G)\leq\frac{4c-4}{c+1},$$

then $\chi_o(G)\leq c$. Cranston proved this conjecture for $c\in\{5,6\}$, while the cases $c\geq 4$ in general remain open. It gives a sharp-looking sparsity threshold guaranteeing an odd coloring.