  We provide evidence that the approach of [Ilyashenko 1991] to the proof of
Dulac's theorem has a gap. Although the asymptotics of [Ilyashenko 1991]
capture far more than the asymptotics of Dulac, we prove that the arguments for
why the asymptotics in [Ilyashenko 1991] are not themselves oscillatory is
insufficient. We give an explicit counterexample and we draw confines to which
Ilyashenko's result may be restricted in order to keep the validity.
