The Graph Complement Conjecture for the Colin de Verdière parameter $\nu$

Let $G$ be a graph, let $G^c$ be its complement, and let $\nu(G)$ be the maximum nullity among positive semidefinite matrices in $\mathcal{S}(G)$ satisfying the Strong Arnold property. Let $\ord G$ denote the number of vertices of $G$.

Graph Complement Conjecture for $\nu$. For any graph $G$,

$$\nu(G)+\nu(G^c)\geq \ord G-2.$$

The inequality is stronger than the corresponding minimum-rank conjectures because $\nu(G)\leq \operatorname{M}_+(G)\leq \operatorname{M}(G)$. The source gives no general resolution here, although later results in the paper establish conditional bounds.