Laplacian spread conjecture for graphs

Let $G$ be a simple graph of order $n\ge 2$, with Laplacian eigenvalues

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

The Laplacian spread conjecture. The inequality

$$\lambda_n(G)-\lambda_2(G)\le n-1$$

holds, or equivalently

$$\lambda_2(G)+\lambda_2(\overline G)\ge 1.$$

Moreover, equality holds if and only if $G$ or $\overline G$ is isomorphic to the join of an isolated vertex and a disconnected graph of order $n-1$. The conjecture concerns the Laplacian spread, an extremal spectral invariant of a graph and its complement. The supplied text does not state whether the conjecture has been resolved.