The Alvarez et al. missing-piece conjecture for locally discrete circle groups

Let $G\subset \operatorname{Diff}_+^\omega(\mathbb S^1)$ be a locally discrete, finitely generated subgroup. The dichotomy in the classification theorem stated immediately before this claim holds for every such group $G$. Here the dichotomy is the alternative between the virtually free case with nonempty exceptional set and the finite-central-extension-of-a-cocompact-Fuchsian-group case, subject to the hypotheses stated in that theorem. This conjecture concerns the missing case in the classification of locally discrete real-analytic circle groups; the source does not provide evidence of resolution.