Non-contractibility below the diameter scale

Determine whether VR(T_{n,n},k) is non-contractible for every scale k below n when n is even and below n−1 when n is odd.

Background

The paper proves contractibility once the scale reaches the diameter of the torus grid: n for even n and n−1 for odd n. The final question asks whether these thresholds are sharp, meaning that no smaller scale ever produces a contractible complex.