Betti-four pattern for the 7+5k family

Determine whether, for every k≥0, the complex VR(T_{7+5k,7+5k},3+2k) has fourth Betti number 14+10k.

Background

Computations confirm the proposed formula for 0≤k≤4. The open problem asks whether the observed arithmetic progression in beta_4 continues throughout the entire infinite family of parameter pairs.

References

Is it true that for $k\geq 0$, the complex $VR{T_{7+5k,7+5k}{3+2k}$ has $\beta_4(VR{T_{7+5k,7+5k}{3+2k})=14+10k$?

— Vietoris-Rips complexes of torus grids  (2502.07134 - Adams et al., 10 Feb 2025) in Question 11, Section Conclusion