Given a 3-manifold with toric boundary, the Culler-Shalen theory associates an incompressible surface to any ideal point of its character variety. From the proof of the Neuwirth conjecture, one knows that necessarily one of those surfaces is a separating surface, hence non Seifert. Is there a way to characterize the case when no ideal point in an component that contains an irreducible character gives rise to a Seifert surface? In the paper "Twisted Alexander Polynomial of Hyperbolic knots", Dunfield, Friedl and Jackson provide examples of detected Seifert surfaces, and state that no fibered 3-manifold's character variety can hold such an ideal point. Is there a more general statement?