Bubbling of almost critical points of anisotropic isoperimetric problems with degenerating ellipticity

Given a sequence of uniformly convex norms ${\phi }_h$ on ${\mathbf{R}}^{n+1}$ converging to an arbitrary norm $\phi$, we prove rigidity of $L^1$-accumulation points of sequences of sets $E_h\subseteq {\mathbf{R}}^{n+1}$ of finite perimeter, that are volume-constrained almost-critical points of the anisotropic surface energy functionals associated with ${\phi }_h$. Here, almost criticality is measured in terms of the $L^n$-deviation from being constant of the distributional anisotropic mean ${\phi }_h$-curvature of (the varifold associated to) of the reduced boundaries of $E_h$. We prove that such limits are finite union of disjoint, but possibly mutually tangent, $\phi$-Wulff shapes.