Uniqueness conjecture for small-impulse Sadovskii vortex patches

Let $S_{\mu,1,1}$ denote the set of maximizers associated with impulse $\mu$, and let $\mu$ be the impulse parameter. Small-impulse uniqueness conjecture. There exists a universal constant $\mu^*>0$ such that, for all $0<\mu\leq\mu^*$, the set of maximizers $S_{\mu,1,1}$ coincides with the $x_1$-translations and rescalings of a single patch, which may then be called the Sadovskii vortex patch. Furthermore, for all $\mu>\mu^*$, $S_{\mu,1,1}$ contains only patches separated from the horizontal axis. This conjecture would yield uniqueness, and hence stability under symmetric perturbations, for sufficiently small impulse, while describing a transition to patches detached from the horizontal axis for larger impulse.