Uniqueness conjecture for minimizers in the defocusing antiperiodic problem

Let $b<0$, and consider the minimization problem in $A_{T/2}$ with fixed mass $m$, denoted by

. For $\alpha,\beta>0$ and $k\in(0,1)$, define the rescaled Jacobi elliptic sine by

\operatorname{sn}_{\alpha,\beta,k}(x)=\frac{1}{\alpha}\operatorname{sn}\left(\frac{x}{\beta},k\right).

$$**Uniqueness conjecture.** The unique minimizer of$$

, up to translations and phase shifts, is $\operatorname{sn}_{\alpha,\beta,k}$, where $\alpha$, $\beta$, and $k$ are uniquely determined. In particular, if $k\in(0,1)$, $\operatorname{sn}(x,k)$ is written as $\operatorname{sn}$, $b=-2k^2$, $T=4K(k)$, and $m=\mathcal M(\operatorname{sn})$, then the unique minimizer, up to translations and phase shifts, is $\operatorname{sn}$. This conjecture identifies the elliptic-function minimizer in the defocusing case and asserts uniqueness of all scaling parameters; the supplied text gives no evidence that it has been resolved.