Homology of the 3k-2 diagonal family

Determine whether, for every k≥3, the reduced homology of VR(T_{3k-2,3k-2},k) is nonzero exactly in dimensions 3 and 4, and establish the stated parity-dependent formulas for its third and fourth Betti numbers for k≥5.

Background

Earlier in the paper, the authors state a computational conjecture for the family with grid size 3k−2 and scale k. The conclusion refines this conjecture by asking both for the support of reduced homology and for explicit formulas for beta_3 and beta_4 depending on the parity of k.

References

For $k\ge 3$, is it the case that $\tilde{H}i(VR{T{3k-2,3k-2}{k})$ is nonzero if and only if $i=3,4$?

For $k\ge 5$, is \beta_3(VR{T_{3k-2,3k-2}{k})= \begin{cases} 6k-3&\text{if }k\text{ is odd} \ 6k-1&\text{if }k\text{ is even?} \end{cases}

— Vietoris-Rips complexes of torus grids  (2502.07134 - Adams et al., 10 Feb 2025) in Question 4, Section Conclusion