Chromatic-number-four unresolved case

Determine whether the principal inertia inequality 2n^+(G) ≤ n^-(G)(n^-(G)+1) holds for graphs with chromatic number 4 and exactly four negative adjacency eigenvalues.

Background

Using a fractional-chromatic-number bound, the authors establish the principal inertia conjecture for graphs with chromatic number at most 3 and for chromatic-number-four graphs having at least five negative eigenvalues.

The only remaining case in that argument is a graph with chromatic number 4 and n^−(G)=4. The authors then prove the conjecture for two subclasses of this case—graphs of maximum degree 4 and planar graphs—but leave the general chromatic-number-four case unresolved.