An explicit counterexample to the equivariant $K=2$ conjecture

We construct an explicit example of a geometrically finite Kleinian group $G$ with invariant component $\Omega$ in the Riemann sphere ${\bf\hat{C}}$ such that any quasiconformal map from $\Omega$ to the boundary of the convex hull of ${\bf\hat{C}} - \Omega$ in ${\bf H^3}$ which extends to the identity map on their common boundary in ${\bf\hat{C}}$, and which is equivariant under the group of Möbius transformations preserving $\Omega$, must have maximal dilatation $K > 2.002$.