Maggi's conjecture on anisotropic critical sets

Let $H$ be a positive convex integrand. A Wulff shape is the set associated with $H$, and consider sets of finite perimeter and finite volume that are critical points of the anisotropic boundary energy under fixed volume constraints.

Maggi's conjecture. Wulff shapes are the unique such sets.

This conjecture generalizes the classical Alexandrov theorem and its anisotropic counterpart. It is known in full generality for the standard Euclidean norm and for elliptic integrands of class $C^{2,\,\alpha}$ with $\alpha>0$, while the general case remains open.