Multiple minimizers via symmetry for a three-fold symmetric elliptic system

Let $\Omega=B_1(0)\subset\mathbb{R}^2$ be the unit disk, and impose the boundary conditions from Example, namely

$$\phi_1(\theta)=\max\{\cos(3\theta),0\},\qquad \phi_2(\theta)=\max\{\cos(3\theta-2\pi/3),0\},\qquad \phi_3(\theta)=\max\{\cos(3\theta-4\pi/3),0\}.$$

Multiple minimizers via symmetry. The constrained problem admits at least three distinct minimizers related by rotations through $2\pi/3$.

This predicts nonuniqueness caused by the three-fold symmetry of the domain and boundary data; the supplied text does not establish whether the claim has been proved or remains open.