  We study the Renormalisation Group (RG) running of the non-singlet tensor
operator, for $N_\mathrm{\scriptstyle f}=3$ QCD with Wilson fermions in a mixed
action setup, with standard Schr\"odinger Functional (SF) boundary conditions
for sea quarks and chirally rotated Schr\"odinger Functional ($\chi$SF)
boundary conditions for valence quarks. Based on a recursive finite-size
scaling technique we compute non-perturbatively the tensor step-scaling
function for an energy range between a hadronic scale and an electroweak scale,
above which perturbation theory may be safely applied. Our result is expressed
as the RG-running factor $T^{\mathrm{RGI}}/[
T(\mu_{\mathrm{had}})]_{\scriptstyle \rm R}$, where the numerator is the scale
independent (Renormalisation Group Invariant - RGI) tensor operator and the
denominator is its renormalised counterpart at a hadronic scale
$\mu_{\mathrm{had}} = 233(8)$~MeV in a given scheme. We determine the
step-scaling function in four distinct renormalisation schemes. We also compute
the renormalisation parameters of these schemes at $\mu_{\mathrm{had}}$ which,
combined with the RG-running factor, gives the scheme-independent quantity
$Z^{\mathrm{RGI}}_{\mathrm T}(g_0^2)$ in four schemes and for a range of bare
gauge couplings in which large volume hadronic matrix element simulations are
performed by the CLS consortium in $N_\mathrm{\scriptstyle f}=2+1$ QCD. All
four results are compatible and also agree with a recent determination based on
a unitary setup for Wilson quarks with Schr\"odinger Functional boundary
conditions~arXiv:2309.04314 . This provides a strong universality test.
