Homogeneity of profinitely rigid crystallographic groups

Determine whether every profinitely rigid crystallographic group is homogeneous, i.e., whether profinite rigidity implies homogeneity for finitely generated virtually abelian groups without non-trivial finite normal subgroups.

Background

The authors construct split crystallographic groups that are not homogeneous and note that these examples are non–profinitely rigid. They ask whether imposing profinite rigidity restores homogeneity. Profinite rigidity connects first-order properties to profinite completions in virtually abelian contexts, and affine Coxeter groups are shown to be profinitely rigid and homogeneous, motivating this question.

Resolving this would bridge model-theoretic homogeneity and profinite properties within the class of crystallographic groups, clarifying how strong rigidity constraints affect definable types.