Laplacian Nordhaus–Gaddum product 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 $\vee$. Laplacian product conjecture.

$$\mu_1(G)\mu_1(\overline G)\leq n(n-1),$$

and equality holds if and only if $G$ or $\overline G$ is isomorphic to $K_1\vee H$ for a disconnected graph $H$ of order $n-1$. The inequality is proved in the source for bipartite graphs, while the general assertion and its equality characterization are left open.