Uniqueness conjecture for closed zero-energy minimizers in the Heisenberg group

Let $\mathcal{H}_1$ be the three-dimensional Heisenberg group, and let $E_1$ be the energy functional considered in the source. For ${\Greekmath 011A}_0>0$, consider the shifted Heisenberg spheres

$$(r^2+\frac{\sqrt{3}}{2}{\Greekmath 011A}_0^2)^2+4t^2={\Greekmath 011A}_0^4.$$

Uniqueness conjecture. The shifted Heisenberg spheres above are the only closed minimizers for $E_1$ with zero energy in $\mathcal{H}_1$. The source establishes that these spheres are closed minimizers with zero energy; the uniqueness among all closed minimizers is the conjectural part.