Circular-arc conjecture for minimizers of the Gamma-limit functional

Assume that $\mathcal{K}$ is not central. Let $M ein\mathbb{N}$ be as in the symmetry condition, and let $(\Delta\theta)_{\min}$ be the minimum total angle compatible with $\mathcal{K}$. Let $v_I^*\in\overline{\mathcal{K}}$ be a minimizer of the \Gamma-limit functional $\overline{\widetilde{\mathcal{A}}_0^\alpha}$. Circular-arc conjecture. If $(\Delta\theta)_{\min}/M<2\pi$, then the trajectory $v_I^*([0,T])$ is composed of circular arcs centered at the origin and passing through some rotation axes, which are swept with uniform motion. The conjecture predicts the structure of minimizers in the noncentral-cone case for $\alpha\in(1,2)$; the supplied text does not state whether it has been proved or disproved.