SE(s) extremal conjecture for weighted Laplacian spread

Let $G$ be a weighted graph on $n\geq 5$ vertices, with every edge weight in $[0,1]$, and let $G^c$ be its weighted complement. Write

$$x=\lambda_2(G),\qquad y=\lambda_2(G^c).$$

An SE$(s)$ graph is the weighted graph family defined earlier in the paper.

SE$(s)$ extremal conjecture. The quantities $x$ and $y$ satisfy

$$x+y-\frac{2xy}{n}\geq 1,$$

with equality when $G$ is an SE$(s)$ graph. The conjecture proposes that the SE$(s)$ family gives the extreme values of Laplacian spread among weighted graphs with non-negative weights whose complements also have non-negative weights; its general status is unresolved.