Aouchiche–Hansen signless Laplacian Nordhaus–Gaddum product conjecture

Let $G$ be a simple graph on $n\geq 2$ vertices, let $\overline G$ be its complement, and let $q_1(H)$ denote the largest signless Laplacian eigenvalue of a graph $H$. Aouchiche–Hansen product conjecture.

$$q_1(G)\\,q_1(\overline G)\leq 2n(n-2).$$

Equality holds if and only if $G$ is the star $K_{1,n-1}$. The source presents this as an open conjecture of Aouchiche and Hansen.