The normalized Laplacian maximum-connectivity conjecture

Let $G$ be a graph on $n\geq2$ vertices, and let $\mathcal{L}(G)$ be its normalized Laplacian matrix with eigenvalues

$$0=\lambda_1(G)\leq\lambda_2(G)\leq\cdots\leq\lambda_n(G).$$

Normalized Laplacian maximum-connectivity conjecture.

$$\max\{\lambda_2(G),\lambda_2(G^c)\}\geq \frac{2}{n-1}.$$

This is a normalized-Laplacian analogue of the Laplacian spread problem and concerns how well-connected a graph or its complement must be. Numerical computations motivate the bound, which remains open.