McMullen's Carathéodory hyperbolicity conjecture for rational-map moduli spaces

Let $f$ be a rational map of degree $d\geq 2$ on the Riemann sphere, and let $\mathcal{M}_f$ be its moduli space of quasiconformal conjugacy classes. A rational map is flexible Lattès if it belongs to the flexible Lattès family. McMullen's conjecture. If $f$ is not flexible Lattès, then $\mathcal{M}_f$ is Carathéodory hyperbolic, meaning that bounded holomorphic functions separate points of $\mathcal{M}_f$. The conjecture predicts strong function-theoretic hyperbolicity for moduli spaces of rational maps; the paper's abstract states that it is solved by constructing a normal affine variety containing a precompact holomorphic image of $\mathcal{M}_f$.