Homotopy types at scale four

Determine the homotopy types of VR(T_{6,6},4), VR(T_{7,7},4), VR(T_{8,8},4), VR(T_{9,9},4), and VR(T_{10,10},4), including whether VR(T_{7,7},4) is homotopy equivalent to a 3-sphere.

Background

The paper proves that VR(T_{7,7},4) is homotopy equivalent to S3, but presents the broader family of scale-four complexes as an open direction. The question records the remaining classifications requested by the authors and explicitly revisits the 3-sphere phenomenon at (n,k)=(7,4).

References

What are the homotopy types of $VR{T_{6,6}{4}$, $VR{T_{7,7}{4}$, $VR{T_{8,8}{4}$, $VR{T_{9,9}{4}$, and $VR{T_{10,10}{4}$? In particular, is $VR{T_{7,7}{4}$ homotopy equivalent to a $3$-sphere?

— Vietoris-Rips complexes of torus grids  (2502.07134 - Adams et al., 10 Feb 2025) in Question 2, Section Conclusion