Brondani–de Lima–Oliveira conjecture on the second signless Laplacian eigenvalues

Let $G$ be a graph on $n$ vertices, and let $\lambda_2(G)$ denote its second signless Laplacian eigenvalue. Brondani–de Lima–Oliveira conjecture.

$$\lambda_2(G)+\lambda_2(\overline{G})\leq -1+\sqrt{\frac{n^2}{2}-n+1}.$$

This conjecture slightly improves the previously known bound of Nikiforov and Yuan. It has been confirmed for trees, $k$-cyclic graphs, regular bipartite graphs, complete multipartite graphs, generalized line graphs and exceptional graphs, but is not resolved in general.