Choi–Greene–Lee–Marquis local surjectivity conjecture for weakly orderable orbifolds

Let $P$ be a polyhedron with a set of vertices $\mathcal{V}$, and suppose that $P$ is weakly $\mathcal{V}$-orderable. Let $\mathcal{O}$ be a Coxeter orbifold structure on $P$ with an ideal or hyperideal end structure, and let $\partial_{\mathcal{V}}\mathcal{O}$ be the disjoint union of the end orbifolds corresponding to $\mathcal{V}$. Let $\mathfrak{D}(\mathcal{O})$ and $\mathfrak{D}(\partial_{\mathcal{V}}\mathcal{O})$ denote the corresponding deformation spaces, and let the hyperbolic point be the point determined by the hyperbolic structure. Choi–Greene–Lee–Marquis local surjectivity conjecture. The function

$$\mathfrak{D}(\mathcal{O})\longrightarrow\mathfrak{D}(\partial_{\mathcal{V}}\mathcal{O})$$

is locally surjective at the hyperbolic point. This is a local version of the preceding end-mapping conjecture, asserting that nearby end deformations are induced by nearby global deformations; no resolution is given in the supplied text.