Uniqueness conjecture for Trudinger–Moser maximizers

Let $\Omega\subset\mathbb{R}^n$ be a ball, and consider the Trudinger–Moser variational problem

$$A=\sup_{\substack{u\in W^{1,n}_0(\Omega)\\ \|u\|\leq 1}}\int_\Omega e^{n\omega_n^{\frac{1}{n-1}}|u|^{\frac{n}{n-1}}}.$$

A maximizer is a function attaining this supremum. Uniqueness conjecture. Maximizers are unique. The preceding discussion recalls that the supremum is attained by a positive radial function on a ball, but the statement does not specify the sense in which uniqueness is intended, such as uniqueness up to sign or other symmetries. The resolution status is not supplied.