  We study the chord-arc Jordan curves that satisfy the Cotlar-type inequality
$T_*(f)\lesssim M^2(Tf),$ where $T$ is the Cauchy transform, $T_*$ is the
maximal Cauchy transform and $M$ is the Hardy-Littlewood maximal function.
Under the background assumption of asymptotic quasi-conformality we find a
characterization of such curves in terms of the smoothness of a parametrization
of the curve.
