Radial minimizer conjecture for the total combined energy

Let $\mathbb{A}$ and $\mathbb{B}$ be the annular domains appearing in the class $\mathcal{F}(\mathbb{A},\mathbb{B})$ of admissible mappings, and let $\mathfrak{E}:\mathcal{F}(\mathbb{A},\mathbb{B})\to\mathbf{R}$ be the total combined energy integral. A mapping is radial if it has the form determined by the radii of the annuli, namely $h(z)=H(|z|)z/|z|$ for a suitable function $H$. Radial minimizer conjecture. The integral $\mathfrak{E}$ attains its minimum in $\mathcal{F}(\mathbb{A},\mathbb{B})$ at a radial mapping $h_\circ$, without assuming any convexity hypothesis. The conjecture asserts that the convexity assumption used in the preceding minimization theorem is not essential; whether radial minimizers always exist under the remaining hypotheses is left open.