Complementary-dimension bounds conjecture

Let $G$ be a graph, let $\overline G$ be its complement, and let $\rho ^\perp$ denote orthogonal representation dimension. The complementary-dimension bounds conjecture. The two dimensions satisfy

$$\rho ^\perp(G)+\rho ^\perp(\overline G)\ge |V(G)|-2$$

and

$$\rho ^\perp(G)+\rho ^\perp(\overline G)\le |V(G)|+2.$$

The source states that these bounds are known for cycles and their complements, but otherwise says that they have not been proved; it also notes that the second bound is false for the general-position parameter.