Betti numbers for the 3k-3 diagonal family

Determine whether, for every k≥7, VR(T_{3k-3,3k-3},k) has beta_3=3 and beta_4=2.

Background

The computational data suggest a stable homological pattern for the family whose grid size is 3k−3 and whose scale is k. The authors ask whether the observed third- and fourth-dimensional Betti numbers persist for all k≥7.

References

Is it true that for $k\geq 7$, the complex $VR{T_{3k-3,3k-3}{k}$ has

\beta_3(VR{T_{3k-3,3k-3}{k})=3 \quad\text{and}\quad \beta_4(VR{T_{3k-3,3k-3}{k})=2?

— Vietoris-Rips complexes of torus grids  (2502.07134 - Adams et al., 10 Feb 2025) in Question 5, Section Conclusion