  We present preliminary results of the non-perturbative renormalization group
(RG) running of the flavor non-singlet tensor operator. We employ the $\chi$SF
scheme for $N_f=3$ QCD using ensembles generated by the ALPHA collaboration for
the computation of the quark mass running. The $\chi$SF property of automatic
$O(a)$ improvement prevents the $O(a)$ mixing of the correlation functions.
