Zhai–Shu–Hong conjecture on the Laplacian spread

Let $G$ be a graph of order $n\geq 2$, and let $\mu_1(G)\geq\cdots\geq\mu_{n-1}(G)$ be its Laplacian eigenvalues. The Laplacian spread is $\mu_1(G)-\mu_{n-1}(G)$. Zhai–Shu–Hong conjecture.

$$\mu_1(G)-\mu_{n-1}(G)\leq n-1,$$

with equality if and only if $G$ or $\overline{G}$ is isomorphic to the join of an isolated vertex and a disconnected graph of order $n-1$.

This is a Nordhaus–Gaddum-type spectral conjecture about the extremal spread of the Laplacian eigenvalues. The source attributes it to Zhai, Shu and Hong, with related attribution to You and Liu, and does not state a resolution.