Let $S_{+}(G)$ denote the sum of the squares of the positive eigenvalues of the adjacency matrix of a graph $G$. Let $S_{-}(G)$ denote the sum of the squares of the negative eigenvalues and $q$ the chromatic number. Is it true that $1 + \frac{S_{+}(G)}{S_{-}(G)} \leq q$?

The chromatic number cannot be replaced with the clique number in this conjecture, because the Coxeter graph provides a counter-example. This bound is incomparable with the Hoffman lower bound for the chromatic number.

UPDATE: Ando and Lin have proved this conjecture in "Proof of a conjectured lower bound on the chromatic number of a graph", to be published soon in Linear Algebra and its Applications. They also prove that $1 + \frac{S_{-}(G)}{S_{+}(G)} \leq q$.