Oscillation conjecture for recurrence sequences without positive dominating roots

Let $(a(n))$ be a recurrence sequence, and let its dominating characteristic roots be the characteristic roots of maximal modulus among those occurring with nonzero coefficient in its generalized power-sum representation. Oscillation conjecture. If $(a(n))$ has no real positive dominating characteristic root, then there are infinitely many $n$ with $a(n)>0$ and infinitely many $n$ with $a(n)<0$. The conjecture predicts that the absence of a positive real characteristic root of maximal modulus forces a recurrence sequence to change sign infinitely often; the surrounding discussion motivates this through oscillating terms from non-real dominating roots, but the supplied text does not indicate whether the conjecture has been resolved.