Uniqueness conjecture for Trudinger–Moser maximizers on balls

Let $\alpha=4\pi$ and let $\Omega$ be a ball. Consider the Trudinger–Moser inequality on $\Omega$, with maximizers taken among the admissible functions in the inequality.

Uniqueness conjecture. If $\alpha=4\pi$ and $\Omega$ is a ball, then the maximizers of the Trudinger–Moser inequality are unique.

Carleson and Chang proved that the supremum is attained by a radial maximizer when $\alpha=4\pi$ and $\Omega$ is a ball. The conjecture asks whether this maximizer is the only one; the cited context does not state that the conjecture has been resolved.