A counterexample concerning smooth approximation

We answer a question of Smith, Stanoyevitch and Stegenga in the negative by constructing a simply connected planar domain $\Omega$ with no two-sided boundary points and for which every point on $\Omega ^c$ is an $m_2$-limit point of $\Omega ^c$ and such that $C^\infty (\overline {\Omega })$ is not dense in the Sobolev space $W^{k,p}(\Omega )$.