Let $\mathbb{D}$ be the unit disk endowed with the Poincaré metric and $G$ be a Fuchsian group such that the hyperbolic surface $\mathbb{D}/G$ is homeomorphic to the plane minus a Cantor set.

**Question: Is there a conformal bijection between $\mathbb{D}/G$ and $\mathbb{C} \setminus K$ for some Cantor set $K \subset \mathbb{C}$?**

I'm pretty sure the answer is yes but the version of the uniformization theorem I know doesn't imply this (just that the plane minus any Cantor set can be uniformized by some Fuchsian group $G$). Any good references?

Thanks in advance!