Anisotropic Serrin-type conjecture for rough domains

Let $H$ be a positive convex integrand, and let a Wulff shape denote the set associated with $H$. Consider indecomposable sets of finite perimeter and finite volume, or alternatively Lipschitz domains, satisfying the overdetermined system referenced in the source.

Anisotropic Serrin-type conjecture for rough domains. Wulff shapes are the unique such sets satisfying the overdetermined system.

This is an anisotropic generalization of Serrin's overdetermined theorem and is connected with Alexandrov's theorem and the one-phase Bernoulli free boundary problem. A key implication would be that the upper bound condition on the perimeter density used in the cited result might be dispensed with; the source explicitly says that this remains open.