Iwaniec's diffeomorphic minimizer conjecture for the disk

Let $f_0:\overline{\mathbb{D}}\to\overline{\mathbb{D}}$ be the prescribed boundary homeomorphism, let $1<p<\infty$, and let $\mathscr{H}^p_{f_0}(\overline{\mathbb{D}},\overline{\mathbb{D}})$ denote the class of finite-distortion homeomorphisms with boundary values $f_0$ and finite energy $\mathscr{E}_p$. Iwaniec's diffeomorphic minimizer conjecture. There exists $f\in\mathscr{H}^p_{f_0}(\overline{\mathbb{D}},\overline{\mathbb{D}})$ such that

$$\mathscr{E}_p[f]=\min_{g\in\mathscr{H}^p_{f_0}(\overline{\mathbb{D}},\overline{\mathbb{D}})}\mathscr{E}_p[g],$$

and this minimizer is a $C^\infty$-smooth diffeomorphism from $\mathbb{D}$ onto $\mathbb{D}$. The conjecture is attributed to Iwaniec and concerns regularity and injectivity of energy minimizers in planar distortion theory; the paper recalls it as unresolved.