Thurston's $K=2$ conjecture for conformal maps to domes

Let $\Omega\subset\mathbb{C}$ be a simply connected domain. Define $K(\Omega)$ as the infimum of the maximal dilatations of quasiconformal maps in $\mathcal{F}$, and $K_{\mathrm{eq}}(\Omega)$ similarly using the equivariant class $\mathcal{F}_{\mathrm{eq}}$. Thurston's $K=2$ conjecture. For any simply connected domain $\Omega\subset\mathbb{C}$,

$$\sup_{\Omega}K(\Omega)=\sup_{\Omega}K_{\mathrm{eq}}(\Omega)=2.$$

The conjecture was refuted by counterexamples in both the equivariant and general cases, so it is no longer open.