The positive semidefinite Graph Complement Conjecture

Let $G$ be a graph, and let $\operatorname{mr}_+(G)$ denote the minimum rank among positive semidefinite matrices in the class $\mathcal{S}(G)$ of real symmetric matrices described by $G$. Write $G^c$ for the complement of $G$ and $\ord G$ for its number of vertices.

Positive semidefinite Graph Complement Conjecture. For any graph $G$,

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

This strengthens the ordinary Graph Complement Conjecture, and the source notes that the bound is sharp for paths; no general resolution is supplied.