Consider a polytope with a $2$-dimensional surface and the corresponding metric on this surface (coming from the embedding in $3$-dimensional Euclidean space). Intrinsically the metric is flat everywhere apart from the vertices of the polytope, where one has cone-like singularities if the angle sum does not equal $2\pi$.

Is every conformal manifold equivalent to such a flat cone-manifold? More precisely, is there a sequence of equivalent conformal manifolds that approximates such a singular manifold?