Smooth manifold structure conjecture for hyperbolic cone metrics

Let $S$ be the marked surface, let $\mathcal{P}$ be its marked points, let $n$ be the relevant parameter, and let $\alpha\in(0,2\pi)^{\mathcal{P}}$ be the tuple of angle defects. For each triangulation $\mathcal{T}$ of $S$, let $\varphi_{\mathcal{T}}$ be the corresponding coordinate map on $\operatorname{Hyp}_{\alpha}(S)$. Smooth manifold structure conjecture. Under some constraint on the cone angles, in particular when

$$\sum_{p\in\mathcal{P}}\alpha_p>2\pi(n-1),$$

the cone angles at the last $n-3$ marked points are near $4\pi$, and the inequality from Corollary~ is satisfied, $\operatorname{Hyp}_{\alpha}(S)$ admits a smooth manifold structure for which every map $\varphi_{\mathcal{T}}$ is a diffeomorphism onto its image. The preceding atlas gives $\operatorname{Hyp}_{\alpha}(S)$ a topological manifold structure, while the implicit-function-theorem argument is expected to establish smoothness for at least some angle data; the precise angle constraints remain to be verified.