The KMT homeomorphism conjecture for Delaunay circle patterns

Let $S_g$ be a closed surface of genus $g$, let $E$ be the edge set of a triangulation of $S_g$, and let $\Theta:E\to[0,\pi)$ be a Delaunay angle structure. Let $P(\Theta)$ be the space of cross ratio systems with prescribed Delaunay angles, and let

$$P(\Theta)\xrightarrow{f}P(S_g)\xrightarrow{\pi}\operatorname{Teich}(S_g)$$

be the forgetful map to the space of marked complex projective structures followed by uniformization. KMT homeomorphism conjecture. For every Delaunay angle structure $\Theta$, the map

$$\pi\circ f:P(\Theta)\to\operatorname{Teich}(S_g)$$

is a homeomorphism. In particular, $P(\Theta)$ is a smooth manifold of real dimension $6g-6$, and $f:P(\Theta)\to P(S_g)$ is an embedding, so that $f(P(\Theta))$ is a section of the bundle $P(S_g)\to\operatorname{Teich}(S_g)$. The conjecture would identify the deformation space of circle patterns with Teichmüller space and would establish the required smoothness; the paper discusses this as an open problem and studies consequences conditional on it.