The finite Riemann surface two-branch-point covering conjecture

A finite Riemann surface $U$ is a Riemann surface whose boundary consists of finitely many analytic Jordan curves. Let $\mathbb{D}$ denote the unit disk, and call a holomorphic map a branched covering if it is a covering map away from its branch points. A map is finite-type when its set of singular values is finite.

Two-branch-point covering conjecture. On every finite Riemann surface $U$, there is a branched covering

$$f\colon U\to\mathbb{D}$$

branched over at most two points. In particular, if $Y$ is any compact hyperbolic surface, and $\pi\colon\mathbb{D}\to Y$ is its universal cover, then

$$\pi\circ f\colon U\to Y$$

is a finite-type map.

This would extend the known constructions of finite-type maps from certain finite Riemann surfaces to compact hyperbolic surfaces to every finite Riemann surface. The statement is presented as plausible in the source, and no proof or resolution is supplied here.