About 5 months ago, I was shown a counterexample to this conjecture by Bojan Mohar (I believe in joint work with two of his grad. students - Javad Ebrahimi and Azhvan Sheikh). However, thanks to Gordon Royle, I have just learned that this problem was resolved earlier by Vladimir Nikiforov. See Linear combinations of graph eigenvalues.
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About 5 months ago, I was shown a counterexample to this conjecture by Bojan Mohar (I believe in joint work with two of his grad. students - Javad Ebrahimi and Azhvan Sheikh). However, thanks to Gordon Royle, I have just learned that this problem was resolved earlier by Vladimir Nikiforov. See Linear combinations of graph eigenvalues.