Infinitesimal rigidity conjecture for cyclic covers of finite-volume hyperbolic manifolds

Let $M$ be a finite-volume hyperbolic manifold of dimension at least $4$. Let $f\colon M\to S^1$ be a non-homotopically trivial smooth map such that $\ker(f_*)$ is finitely generated, where $f_*$ is the map induced on fundamental groups. Let $\widetilde{M}_f$ denote the cyclic covering associated to the subgroup $\ker(f_*)$. Infinitesimal rigidity conjecture. The cyclic covering $\widetilde{M}_f$ is infinitesimally rigid.

The conjecture is motivated by examples in which $\ker(f_*)$ is finitely generated and the authors' method proves, or gives strong numerical evidence for, infinitesimal rigidity of the associated cyclic cover. The general assertion for finite-volume hyperbolic manifolds in dimensions at least $4$ remains open.