Upper vs. lower F-homology growth equality

Determine whether there exists a finite CW complex X and a field F for which the upper and lower F-homology growth invariants differ in some degree k; equivalently, ascertain whether the upper and lower F-homology growth invariants always coincide for every finite CW complex X and field F. Concretely, for the invariants defined by normalized Betti numbers over towers of finite covers (the upper growth given by an infimum of suprema and the lower growth given by a supremum of infima), decide if there is an example with unequal values or prove equality holds universally.

Background

Section 8 introduces the upper and lower F-homology growth invariants for a finite complex X by considering normalized Betti numbers across finite covers. These invariants capture asymptotic homological behavior under coverings and are defined via an infimum of suprema (upper growth) and a supremum of infima (lower growth).

The authors note several basic properties, including homotopy invariance, multiplicativity under finite covers, and additivity over disjoint unions. They also record that over the rationals these invariants agree, and mapping tori have vanishing growth. Despite these facts, it remains unresolved whether the two growth invariants can differ in general for some field F, prompting the explicit remark below.