On a question of T. Sheil-Small regarding valency of harmonic maps

The aim of this work is to answer positively a more general question than the following which is due to T. Sheil-Small: Does the harmonic extension in the open unit disc of a mapping f from the unit circle into itself of the form $f(e^{it})=e^{i\varphi (t)}$, $0\le t\le 2\pi$ where $\varphi$ is a continuously non-decreasing function that satisfies $\varphi (2\pi )-\varphi (0)=2N\pi$, assume every value finitely many times in the disc?