The odd chromatic number conjecture for connected graphs

Let $G$ be a connected graph with maximum degree $\Delta\geq3$. A proper coloring of $G$ is odd if, for every non-isolated vertex $v$, some color appears an odd number of times in the open neighborhood $N(v)$; write $\chi_o(G)$ for the minimum number of colors in an odd coloring.

Odd chromatic number conjecture. If $G$ is a connected graph of maximum degree $\Delta\geq3$, then

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

This conjecture was previously proposed in connection with the odd chromatic number, a parameter satisfying $\chi(G)\leq\chi_o(G)\leq\chi_{\rm pcf}(G)$. Its resolution is not given in the supplied text.