  In 2010, Kerckhoff and Storm discovered a path of hyperbolic 4-polytopes
eventually collapsing to an ideal right-angled cuboctahedron. This is expressed
by a deformation of the inclusion of a discrete reflection group (a
right-angled Coxeter group) in the isometry group of hyperbolic 4-space. More
recently, we have shown that the path of polytopes can be extended to Anti-de
Sitter geometry so as to have geometric transition on a naturally associated
4-orbifold, via a transitional half-pipe structure.
  In this paper, we study the hyperbolic, Anti-de Sitter, and half-pipe
character varieties of Kerckhoff and Storm's right-angled Coxeter group near
each of the found holonomy representations, including a description of the
singularity that appears at the collapse. An essential tool is the study of
some rigidity properties of right-angled cusp groups in dimension four.
