  We study regularization scheme dependence of K\"ahler ($N=2$) supersymmetric
sigma models. At the one-loop order the metric $\beta$ function is the same as
in non-supersymmetric case and coincides with the Ricci tensor. First
correction in MS scheme is known to appear in the fourth loop. We show that for
certain integrable K\"ahler backgrounds, such as complete $T-$dual of
$\eta$-deformed $\mathbb{CP}(n)$ sigma models, there is a scheme in which the
fourth loop contribution vanishes.
