Homeomorphism conjecture for neohookean energy minimizers

Let $p\geqslant 2$ and $q>0$. Let $\mathscr M^p (\overline{\mathbb{X}},\overline{\mathbb{Y}})$ be the class of orientation-preserving monotone mappings $h\colon\overline{\mathbb{X}}\xrightarrow[]{{}_{\!\!\textnormal{onto\,\,}\!\!}}\overline{\mathbb{Y}}$ in $\mathscr{W}^{1,p}(\mathbb{X},\mathbb{C})$, and let $h_\circ$ be a minimizer of the neohookean energy $\mathsf E_q^p$ over this class, so that

$$\mathsf E_q^p[h_\circ]=\inf_{h\in\mathscr M^p(\overline{\mathbb{X}},\overline{\mathbb{Y}})}\mathsf E_q^p[h].$$

Homeomorphism conjecture. Every minimizer $h_\circ\in\mathscr M^p(\overline{\mathbb{X}},\overline{\mathbb{Y}})$ is a homeomorphism.

The conjecture asserts that although monotone Sobolev mappings are used to obtain existence of minimizers when boundary values are free, minimizers do not actually collapse continua and hence remain homeomorphisms. Its status is not resolved in the supplied text.