Laplacian spread equality characterization conjecture

Let $G$ be a graph with $n$ vertices, let $\overline G$ be its complement, and let $K_1$ denote the one-vertex complete graph. The join of graphs is denoted by $\vee$, and a graph is disconnected when it is not connected. Laplacian spread equality conjecture.

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

(or equivalently $\mu_1(G)+\mu_1(\overline G)\leq 2n-1$), with equality if and only if $G$ or $\overline G$ is isomorphic to $K_1\vee H$ for a disconnected graph $H$ of order $n-1$. This is the proposed full form of the Laplacian spread conjecture; the equality characterization remains open in general.