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 $f\colon \mathbb {R}^2\to \mathbb {R}^2$ of finite distortion with $\exp (\lambda K)\in L^1_{\mathrm {loc}}(\mathbb {R}^2)$ 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.