Existence conjecture for energy-minimal diffeomorphisms of multiply connected domains

Let $\Omega$ and $\Omega^*$ be bounded $k$-connected domains in $\mathbb{C}$, where $k\geqslant 2$. Suppose that $\Omega\subset\Omega^*$ and that the inclusion is a homotopy equivalence. Existence conjecture. There exists an energy-minimal diffeomorphism of $\Omega$ onto $\Omega^*$. The conjecture extends the doubly connected existence theorem to higher connectivity. It is stated for all $k\geqslant 2$, and the case $k=2$ is known by the theorem cited in the source; the higher-connectivity cases remain open.