Mappings of finite distortion: Formation of cusps II

For $s>0$ given, we consider a planar domain $\Omega_s$ with a rectifiable boundary but containing a cusp of degree $s$, and show that there is no homeomorphism $f\colon\bR^2\to\mathbb{R}^2$ of finite distortion with $\exp(\lambda K)\in L^1_{\loc}(\mathbb{R}^2)$ so that $f(B)=\Omega_s$ when $\lambda>4/s$ and $B$ is the unit disc. On the other hand, for $\lambda<2/s$ such an $f$ exists. The critical value for $\lambda$ remains open.