Existence of smooth minimisers for the ${\cal A}$-mean distortion energy

Let ${\Bbb D}$ be the unit disk, and let $f_0:\overline{{\Bbb D}}\to\overline{{\Bbb D}}$ be a homeomorphism of finite distortion with

$$\mathsf{E}_{\cal A}(f_0)<\infty.$$

Here ${\cal A}:[1,\infty)\to[1,\infty)$ is convex and increasing, satisfies $p{\cal A}(t)\leq t{\cal A}'(t)$ for some $p>1$, and

$$\mathsf{E}_{\cal A}(f):=\int_{\Bbb D}{\cal A}({\Bbb K}(z,f))\,dz$$

is the ${\cal A}$-mean distortion energy. Smooth minimiser conjecture. In the space of homeomorphic mappings of finite distortion with boundary values $f_0$, there is a minimiser $f$ which is also a smooth diffeomorphism. This conjecture concerns existence and regularity of minimisers for distortion functionals under prescribed boundary values; the supplied source recalls it from earlier work, and no resolution is given here.