  We prove the infinitesimal rigidity of some geometrically infinite hyperbolic
4- and 5-manifolds. These examples arise as infinite cyclic coverings of
finite-volume hyperbolic manifolds obtained by colouring right-angled
polytopes, already described in the papers arXiv:2009.04997 [math.GT] and
arXiv:2105.14795 [math.GT]. The 5-dimensional example is diffeomorphic to $N
\times \mathbb{R}$ for some aspherical 4-manifold $N$ which does not admit any
hyperbolic structure. To this purpose we develop a general strategy to study
the infinitesimal rigidity of cyclic coverings of manifolds obtained by
colouring right-angled polytopes.
