Degeneracy of maximizers for highly singular inverse-power weights

Prove or disprove that, for \(n-2<\alpha<n-1\), the maximizer of \(E_{-\alpha}(\partial\Omega)=\int_{\partial\Omega}|x|^{-\alpha}\,d\mathcal{H}^{n-1}(x)\) among convex bodies in \(\mathbb{R}^n\) with fixed perimeter is the degenerate flat ball \(\Omega=\{(x',0):x'\in\mathbb{R}^{n-1},\ |x'|\le R\}\), with \(R\) chosen to satisfy the perimeter constraint.

Background

The authors establish existence of maximizers for decreasing radial weights and show, for $n-2<\alpha<n-1$, that the origin belongs to the boundary of any maximizer. In dimension two, they prove the conjectured degeneracy under an additional symmetry assumption, with the needle as the optimizer. These results motivate a higher-dimensional conjecture that the optimizer is always a lower-dimensional flat ball for the highly singular range $n-2<\alpha<n-1$. The conjecture is not proved in general.