The Graph Complement Conjecture for minimum rank

Let $G$ be a graph. Write $G^c$ for its complement, $\operatorname{mr}(G)$ for the minimum rank of a real symmetric matrix described by $G$, and $\ord G$ for the number of vertices of $G$.

Graph Complement Conjecture. For any graph $G$,

$$\operatorname{mr}(G)+\operatorname{mr}(G^c)\leq \ord G+2.$$

This is a classical Nordhaus–Gaddum-type conjecture in minimum rank theory; the bound is sharp for paths, but the source gives no resolution in general.