Non-computational proof of the homotopy-through-homology theorem

Develop a proof of the homotopy equivalences VR(T_{5,5},3)≃∨_4 S^9 and VR(T_{7,7},4)≃S^3 that does not rely on integral homology computations.

Background

The paper proves the two homotopy equivalences by computing integral homology and then applying connectivity results together with the Hurewicz and Whitehead theorems. The authors explicitly ask whether an alternative argument can establish the same conclusions without those integral homology calculations.