On spaces of minimal higher topological complexity

Let TC$_n$(X) denote the n-th topological complexity of a topological space X. It is known that TC$_n$(X) does not exceed n-1 for non-contractible X, and so it makes sense to describe spaces X with TC$_n$(X) =n-1. Grant--Lupton--Oprea proved the following: If X is a nilpotent space with TC$_n$(X)=n-1 then X is homotopy equivalent to an odd-dimensional sphere. Here we made an attempt to get rid of nilpotency condition and prove the following: If TC$_n$(X) =n-1 then either X is homotopy equivalent to a sphere of odd dimension or is a homology circle with the infinite cyclic fundamental group.