This is probably trivial but has been bothering me all day.

Suppose $f:\Sigma_g\to \mathbb{S}^2$ is a $g+1$ fold branched conformal map with $\Sigma_g$ a connected genus $g$ surface and $f$ having $4$ branch points (all of order $g+1$).

Is it true that $f:\Sigma_g^* \to (\mathbb{S}^2)^*$ (the cover obtained by removing the branch points) is a regular cover (i.e. so the deck group acts transitively on the fibers of $f$)?

I thought this was obvious but then realized I didn't know how to prove it (admittedly this is a bit outside my area of expertise). Part of the problem is that I don't have a good example of a branched map from a connected surface which is not a regular cover away from the branch points.

**Edit**

Based on some comments of Sebastian. It would be enough for my purposes to prove that the conformal structure of $\Sigma_g$ was uniquely determined by the cross-ratio of the branch points. However, its not clear to me whether this can be shown without knowing that the cover is regular...