Non-monotonicity of homology rank

Characterize when the rank of homology for Vietoris–Rips complexes of torus grids can be non-monotonic as the scale parameter increases.

Background

The paper notes an explicit example in dimension four: H_4(VR(T_{10,10},4)) has rank 60, H_4(VR(T_{10,10},5)) has rank 0, and H_4(VR(T_{10,10},6)) has rank 39. Thus, increasing the scale can cause homology to disappear and later reappear, and the authors ask for a general characterization of when this can occur.