Weighted Laplacian Spread Conjecture

Let $G$ be a weighted graph on $n$ vertices in which every edge weight lies in $[0,1]$, and let $G^c$ be its weighted complement, obtained by replacing each edge weight $w_{ij}$ by $1-w_{ij}$. The algebraic connectivity of a weighted graph is denoted by $\lambda_2$.

Weighted Laplacian Spread Conjecture. One has

$$\lambda_2(G)+\lambda_2(G^c)\geq 1.$$

This is the natural extension of the symmetric formulation of the Laplacian Spread Conjecture from simple graphs to weighted graphs; the paper reports supporting investigations but no resolution.