Generalized KMT conjecture for Delaunay circle patterns

Let $S$ be a closed, oriented surface of genus at least $2$. An admissible weighted graph $(\Gamma,\theta)$ in $S$ is a graph embedded in $S$ that is the 1-skeleton of a cell decomposition, with $\theta:E(\Gamma)\to(0,\pi)$ such that the sum of the edge weights around every face is $2\pi$, while the sum around every contractible cycle not contained in the boundary of a face is strictly greater than $2\pi$. Let $\mathcal C_{\Gamma,\theta}$ be the space of pairs $(\sigma,\mathcal C)$ consisting of a complex projective structure $\sigma$ on $S$ and a circle pattern $\mathcal C$ with nerve $\Gamma$ and intersection angles $\theta$. Generalized KMT conjecture. For every admissible weighted graph $(\Gamma,\theta)$, the forgetful map

$$\mathcal C_{\Gamma,\theta}\to\mathcal T_S$$

is a homeomorphism. This generalizes the circle-packing statement to Delaunay circle patterns with prescribed intersection angles. The conjecture has been proved for tori by Wayne Lam; the supplied source does not assert a result for the stated genus-at-least-two setting.