Random-graph inertia conjecture

Let $G$ be sampled from the binomial random graph $\mathbb G(n,\frac 12)$, and let $A$ be a weighted adjacency matrix of $G$. Write $n_{\geq 0}(A)$ for the number of non-negative eigenvalues of $A$. Random-graph inertia conjecture. With probability $1-o(1)$ as $n\to\infty$, every weighted adjacency matrix $A$ of $G$ satisfies

$$n_{\geq 0}(A)=\Omega\left(\frac{n}{\log n}\right).$$

Since the independence number of $\mathbb G(n,\frac 12)$ is $O(\log n)$ with probability $1-o(1)$, this would show that the inertia bound is far from tight for almost all graphs. The bound would also be best possible up to constants, by the corresponding clique-cover estimate.