Elphick–Wocjan conjecture on the spectral lower bound for clique number

Let $G$ be a graph with $n$ vertices, let $\lambda_1,\dots,\lambda_n$ be the eigenvalues of its adjacency matrix, and let $s^+$ be the sum of the squares of its positive eigenvalues:

$$s^+ = \sum_{\lambda_i>0}\lambda_i^2.$$

Let $\omega$ denote the clique number of $G$. Elphick–Wocjan conjecture. The positive-eigenvalue spectral quantity satisfies

$$\sqrt{s^+} \leq n\left(1-\frac{1}{\omega}\right).$$

Equivalently,

$$\frac{n}{n-\sqrt{s^+}}\leq\omega.$$

This conjecture strengthens Wilf's spectral lower bound on the clique number by replacing the largest adjacency eigenvalue with the square root of the sum of the squares of all positive eigenvalues. It is also stated as the second problem in a survey of open problems in spectral graph theory.