  We show that there exists a planar Jordan domains $\Omega$ with boundary of
Hausdorff dimension $1$ such that, for any conformal maps $\varphi \colon
\mathbb D \to \Omega$, any homeomorphic extension of $\varphi$ or
$\varphi^{-1}$ to the entire plane is not in $W^{1,\,1}_{\rm loc}$ (or even not
in $BV_{\rm loc}$).
