Seven-colour conjecture for odd colouring of toroidal graphs

Let $G$ be a simple graph that embeds in the torus. A proper odd vertex-colouring of $G$ is a proper colouring in which, for every non-isolated vertex $v$, some colour appears an odd number of times in the neighbourhood $N(v)$. Let $\chi_o(G)$ denote the smallest number of colours in a proper odd vertex-colouring of $G$.

Seven-colour conjecture for toroidal graphs. Every toroidal graph $G$ satisfies

$$\chi_o(G) \leq 7.$$

The paper proves the bounds $7 \leq \chi_o(T) \leq 9$ for the torus $T$, so the conjecture would determine the exact value $\chi_o(T)=7$. It remains open in the supplied source.