Consider a unit disk with marked points $z_i$, $i=1, \dots , n$ on its boundary. Let us call this surface $X$. As it is well known, the disk can be equipped with an hyperbolic metric and is then called Poincaré disk. Consider the convex geodesic hull of the points $z_i$, i.e. the subset of the disk surrounded by the geodesics $\gamma_i$ joining consecutive pair points $(z_i, z_i+1)$, together with the geodesics and the points $z_i$ themselves. Let us call this surface $Y$.
Is there a biholomorphic map $\phi : X \to Y$ such that the points $\phi(z_i) = z_i$? In other words, are the surfaces with marked points $(X; z_i)$ and $(Y; z_i)$ isomorphic as Riemann surfaces?