The maximum-degree bound for odd chromatic number

Let $G$ be a connected graph with maximum degree $\Delta$. Maximum-degree odd chromatic number conjecture. If $\Delta\geq3$, then

$$\chi_o(G)\leq\Delta+1.$$

This would extend the preceding bounds for subcubic graphs to all connected graphs of maximum degree at least $3$. The statement is presented as a proposed upper bound; its validity is left open in the source.