Homotopy type of the 2k-by-2k family

Determine whether, for every k≥2, VR(T_{2k,2k},k) is homotopy equivalent to the wedge of S^3 and a 4k-fold wedge of S^{2k−1}.

Background

The proposed family lies in the intermediate-scale regime where the paper observes complicated topology. The conjectured homotopy type combines a 3-sphere with a number of higher-dimensional spheres and is intended to capture a pattern suggested by the computed examples.

References

Is it true that for $k\geq 2$, the complex $VR{T_{2k, 2k}{k}$ is homotopy equivalent to the wedge sum of $S3$ with a $4k$-fold wedge sum of $S{2k-1}$?

— Vietoris-Rips complexes of torus grids  (2502.07134 - Adams et al., 10 Feb 2025) in Question 6, Section Conclusion