The graph complement conjecture for minimum rank and maximum nullity

For a graph $G$ on $n$ vertices, let $S(G)$ be the set of real symmetric matrices whose off-diagonal zero pattern is prescribed by adjacency in $G$. Define $mr(G)$ as the minimum rank of a matrix in $S(G)$ and $M(G)=n-mr(G)$ as the maximum nullity. Graph complement conjecture. For any graph $G$,

$$mr(G)+mr(\overline{G})\leq |G|+2$$

and equivalently

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

This conjecture gives equivalent formulations in terms of minimum rank and maximum nullity for a graph and its complement; the source presents it as a possible future direction, but the supplied text does not establish its resolution.