Unweighted Nordhaus–Gaddum conjecture for graph inertia

Let $G$ be a connected graph on $n$ vertices, and let $\overline{G}$ denote its complement. For a graph $H$, let $A_H$ be its unweighted adjacency matrix, and let $n_{\geq 0}(A_H)$ denote the number of nonnegative eigenvalues of $A_H$.

Unweighted Nordhaus–Gaddum conjecture. One has

$$n_{\geq 0}(A_G)\,n_{\geq 0}(A_{\overline{G}})\geq n.$$

The multiplicative Nordhaus–Gaddum inequality is proposed here for unweighted inertia after the corresponding weighted inequality is shown not to hold in its natural form. The additive inequality is known, but the multiplicative bound remains open according to the source.