Weighted Dirichlet-minimization conjecture for homeomorphisms between spherical rings

Let $\mathbb{A}(r,R)$ and $\mathbb{A}(r_\ast,R_\ast)$ be spherical rings in $\mathbf{R}^n$, and let $\mathcal{F}$ be the family of homeomorphisms between them belonging to $\mathscr{W}^{1,n-1}$. For $f\in\mathcal{F}$, define the weighted Dirichlet integral by

$$\mathscr{F}[f]=\int_{\mathbb{A}(r,R)}\frac{\lVert Df\rVert^{n-1}}{|f|^{n-1}}\,dx,$$

where the weight is $\wp(y)=|y|^{1-n}$. Weighted Dirichlet-minimization conjecture. The integral $\mathscr{F}[f]$ achieves its minimum among $\mathcal{F}$ for generalized-radial diffeomorphisms between the annuli. This predicts that the minimizers of the weighted variational problem are generalized-radial maps.