  We present the preliminary results of the non-perturbative running of the
flavour non-singlet tensor operator in the high-energy range
$2~\rm{GeV}\lesssim \mu\lesssim 128~\rm{GeV}$ in $N_f=3$ massless QCD,
comparing four different definitions of the renormalisation constant. We use
the configuration ensembles of arXiv:1802.05243 and arXiv:1607.06423, subject
to Schr\"odinger functional (SF) boundary conditions, and valence quarks with
chirally rotated Schr\"odinger functional ($\chi$SF) boundary conditions.
Provided that boundary counterterms have been appropriately tuned, this results
in O($a$) improvement of the tensor operator, without the need of a dimension-4
Symanzik counterterm (proportional to $c_T$).
