Kojima–Mizushima–Tan uniqueness conjecture for circle packings

Let $S$ be an oriented closed surface of genus $g\geq 2$, let $\tau$ be a polygonal cell decomposition of $S$, and let $c$ be a complex structure on $S$. A complex projective structure on $S$ has underlying complex structure $c$ if its underlying complex structure is $c$; a circle packing has nerve $\tau$ when its tangency cell decomposition is isotopic to $\tau$. Kojima–Mizushima–Tan conjecture. There is a unique complex projective structure $\sigma$ on $S$ with underlying complex structure $c$ admitting a circle packing with nerve $\tau$. This conjecture predicts existence and uniqueness of the complex projective structure realizing prescribed circle-packing combinatorics over every complex structure. The paper proves a properness result for the corresponding forgetful map, but the stated existence-and-uniqueness claim is not resolved here.