The normalized algebraic-connectivity Nordhaus-Gaddum conjecture

Let $G$ be a graph on $n$ vertices, let $G^c$ be its complement, and let $\lambda_2(G)$ denote the second-smallest eigenvalue of the normalized Laplacian of $G$.

Normalized algebraic-connectivity conjecture.

$$\lambda_2(G)+\lambda_2(G^c)\leq 2-O\left(\frac{1}{\sqrt{n}}\right).$$

The conjecture is motivated by focus graphs, for which the two normalized algebraic connectivities approach $1$ while their sum approaches $2$. The supplied text does not provide a proof, disproof, or further resolution.