Uniqueness conjecture for infinity-harmonic maps to the circle

Let $M$ be the manifold under consideration, and let $u:M\to S^1$ be an $\infty$-harmonic map in a fixed homotopy class. Uniqueness conjecture. The map $u$ is unique up to rotation in $S^1$. The authors state that uniqueness results are known in related settings, but their uniqueness proofs do not carry over to maps into $S^1$; even the corresponding Euclidean annular Dirichlet problem is left as a question.