Suppose I have (roughly speaking) a multivalued meromorphic function $f(z)$ on all of $\mathbb{C}$ that is single-valued and holomorphic on the open unit disc and has some branch points of finite order (but no poles) on the unit circle. Does the Taylor series always converge uniformly to $f(z)$ on the closed unit disc? This seems very likely but I do not think I have ever seen a proof.

I know an argument that works for the simplest possible case, namely the function $f(z)=\sqrt{1-z}$.