Traces and Sobolev extension domains

Assume that $\Omega \subset{\mathbb{R}^n}$ is a bounded domain and its boundary $\partial \Omega$ is $m$-regular, $n-1 \le m <n$. We show that if there exists a bounded trace operator $T:W^{1,p}(\Omega) \to B^{p}_{1-\alpha}(\partial\Omega)$, $1<p<\infty$ and $\alpha = \tfrac{n-m}{p}$, and $(1-\lambda)$-Hölder continuous functions are dense in $W^{1,p}(\Omega)$, $0\le \lambda < n-m$, then the domain $\Omega$ is a $W^{1,p}$-extension domain.