Hatcher and Oertel computed the boundary slopes of essential surfaces of Montesinos knots in this paper. But they do not consider surfaces that do not intersect the boundary of the exterior. An essential surface is an incompressible and $\partial$-incompressible surface.
My question is this: Let $K$ be a Montesinos knot in $S^3$ and $\eta(K)$ is an open neighborhood of $K$. Does every essential surface, that is not isotopic to the boundary, intersect the boundary of $S^3-\eta(K)$?
If $K'$ is a knot such that $S^3-\eta(K')$ is Seifert fibered, this result holds as proved by Zupan in Lemma 3.5
Any references or suggestions are appreciated. Thanks.