Relation between conformal blocks and Belyi functions

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Belyi functions are maps from Riemann surfaces to $C\mathbb{P}^1$ ramified exactly at three points. Four point Virasoro conformal block on a sphere as a function of the cross-ratio $x$  is believed to posses singularities only at three points. My question is whether the generic four-point Virasoro conformal block on a sphere is a Belyi function? I am also quite interested in references where conformal blocks might be analysed from this point of view.

How do you use the 4-point function to construct a map $\Sigma \to \mathbb{CP}^1$?
@RyanThorngren  Well, the 4-pt conformal block, being roughly the holomorphic part of the 4-pt correlation function, as a function of the projective invariant $x$ lives on a 3-punctured sphere (Riemann surface) and takes values in complex numbers ($\mathbb{CP^1}$).
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