Nordhaus–Gaddum conjecture for independent position chromatic numbers

Let $G$ be a graph of order $n$, let $\overline{G}$ be its complement, and choose $\pi\in\{\operatorname{gp},\operatorname{mp}\}$, where $\operatorname{gp}$ and $\operatorname{mp}$ denote general and monophonic position, respectively. Let $\chi_{\pi_{\rm i}}(G)$ denote the chromatic number whose color classes are independent $\pi$-position sets. Nordhaus–Gaddum conjecture.

$$\chi_{\pi_{\rm i}}(G)+\chi_{\pi_{\rm i}}(\overline{G})\leq n+1.$$

The survey identifies this as an open problem; no general proof or counterexample is reported.