Axial symmetry conjecture for minimizing disc configurations with elastic boundaries

Let $E$ be the elastic surface energy for topological discs, with positive parameters $\alpha$, $\beta$, $c_o$ and $b\neq 0$, and define

$$\underline E:=2\sqrt{\alpha\,\beta}-\lvert b\rvert.$$

Assume that $\underline E\geq 0$. Axial symmetry conjecture. The infimum of the energy among all topological discs is attained by an axially symmetric surface with nonconstant mean curvature. The preceding result rules out constant-mean-curvature critical discs when $b\neq 0$ and $c_o\neq 0$; the conjecture proposes existence of a minimizer with axial symmetry and nonconstant mean curvature in the stated parameter regime.