Prescribed curvature problem for discrete conformality on convex spherical cone-metrics

Let $S$ be the 2-sphere and $V\subset S$ be a finite set of at least three points. We show that for each function $\kappa :V\to (0,2\pi )$ satisfying elementary necessary conditions, in each discrete conformal class of spherical cone-metrics there exists a unique metric realizing $\kappa$ as its discrete curvature. This can be seen as a discrete version of a result of Luo and Tian.