The Colin de Verdière graph complement conjecture

Let $G$ be a finite simple graph and let $\overline{G}$ denote its complement. Let $\mu(G)$ denote the Colin de Verdière parameter. Colin de Verdière's graph complement conjecture. Every graph $G$ satisfies

$$\mu(G)+\mu(\overline{G})\geq |V(G)|-2.$$

This is the complement analogue of Nordhaus–Gaddum-type inequalities for graph parameters and is related in the paper to the extremal edge conjecture for the Colin de Verdière parameter. The source presents it as a conjecture; no resolution is supplied there.