On beta-function of $N=2$ supersymmetric integrable sigma models

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