The character-variety finiteness conjecture for profinite distinguishability of Dehn fillings

Let $M$ be a one-cusped, finite-volume, hyperbolic $3$-manifold, and let $M_{m/n}$ be a hyperbolic Dehn filling with surgery coefficient $m/n$. Write $\Gamma=\pi_1(M_{m/n})$, and let $\chi^I_\mathbb{C}(\Gamma)$ denote its character variety. Character-variety finiteness conjecture. In the main theorem, the assumption that $\chi^I_\mathbb{C}(\Gamma)$ is finite can be dropped, and consequently the implicit assumption that $M_{m/n}$ is non-Haken can also be dropped. The proposed extension would establish profinite distinguishability for a broader class of hyperbolic Dehn fillings; the source describes a possible approach using zero-dimensional components over $\overline{\mathbb{Q}_p}$, but does not report a proof.