The equality characterization for the normalized Laplacian maximum bound

Let $G$ be a graph on $n\geq5$ vertices, with normalized Laplacian eigenvalues $0=\lambda_1(G)\leq\lambda_2(G)\leq\cdots\leq\lambda_n(G)$. Normalized Laplacian maximum-bound conjecture.

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

with equality if and only if $n$ is odd and $G$ is the join of an isolated vertex with two complete graphs, each of order $\frac{n-1}{2}$. This conjecture refines the proposed lower bound by specifying all equality cases. The paper presents it as an open problem supported by computations and identifies the displayed graph family for odd orders.