Open question: Minimal TC^n with π1(X) ≅ Z (n ≥ 3) implies homotopy circle?

Determine whether a connected CW space of finite type X with fundamental group π1(X) ≅ Z and nth topological complexity TC^n(X) = n − 1 for n ≥ 3 is homotopy equivalent to the circle S^1.

Background

The paper proves that if TC^n(X) = n − 1, then X is either homotopy equivalent to an odd-dimensional sphere or is an integral homology circle with π1(X) ≅ Z. This reduces the classification to determining whether the latter case is actually homotopy equivalent to S^1.

Resolving this question would settle the conjectured classification by confirming that all spaces with minimal higher topological complexity are odd spheres (including S^1).