Maximality of persistent third homology

Determine whether, for every n and k≤k′, the rank of the inclusion-induced map H_3(VR(T_{n,n},k))→H_3(VR(T_{n,n},k′)) equals the minimum of the two third Betti numbers.

Background

The question concerns the behavior of third-dimensional persistent homology under the inclusions induced by increasing the scale. It asks whether these maps retain as much third homology as is permitted by the dimensions of the source and target groups.

References

For any $n$ and for any $k\le k'$, is the rank of the map $H_3(VR{T_{n,n}{k})\to H_3(VR{T_{n,n}{k'})$ induced by inclusion equal to

\min{\beta_3(VR{T_{n,n}{k}),\beta_3(VR{T_{n,n}{k'})}? In other words, are the $3$-dimensional persistent homology groups as large as they can be (given the $3$-dimensional Betti numbers)?

— Vietoris-Rips complexes of torus grids  (2502.07134 - Adams et al., 10 Feb 2025) in Question 9, Section Conclusion