Kähler manifolds, topological rigidity, and finite quotients of groups

I will mention a work in progress theorem of myself, Claudio Llosa Isenrich, Pierre Py, Ryan Spitler, Matthew Stover, and Stefano Vidussi.
The result relates to three seemingly orthogonal rigidity programs:
rigidity of aspherical manifolds (e.g. the Borel Conjecture), profinite rigidity of finitely generated residually finite groups (e.g. the Remeslennikov Conjecture), and (loosely) Grothendieck's anabelian geometry.