Sharpness conjecture for odd colorings of sparse graphs

Let $\mathrm{mad}(G)$ denote the maximum average degree of $G$, and let $\chi_o(G)$ be the smallest number of colors admitting an odd coloring of $G$. Fix $\epsilon$ such that $0<\epsilon\le 8/5$. Sharpness conjecture. If

$$\mathrm{mad}(G)\le 4-\epsilon,$$

then

$$\chi_o(G)\le \left\lfloor\frac{8}{\epsilon}\right\rfloor-1.$$

The source says that this construction is suspected to be sharp, but supplies no resolution evidence; the claim is therefore recorded as open.