Cranston's sparse-graph odd coloring conjecture

Let $G$ be a finite simple graph, let $c\geq 4$, and let $\operatorname{mad}(G)$ be its maximum average degree, defined by

$$\operatorname{mad}(G)=\max_{\emptyset\neq H\subseteq G}\left\{\frac{2|E(H)|}{|V(H)|}\right\}.$$

An odd $c$-coloring is a proper $c$-coloring in which every non-isolated vertex has a color appearing an odd number of times in its open neighborhood, and let $\chi_o(G)$ denote the minimum $c$ for which $G$ has an odd $c$-coloring. Cranston's conjecture. If

$$\operatorname{mad}(G)<\frac{4c}{c+2},$$

then $\chi_o(G)\leq c$. The conjecture is known to hold for $c\in\{5,6\}$, while the $c=4$ formulation is false: a graph whose every block is a $5$-cycle is a counterexample.