  Matching Non-Perturbative Renormalisation on the lattice and perturbative
renormalisation would benefit from higher matching scales, which are needed for
observables entering their permile era such as BK. In this work we lay down a
strategy, within the Rome-Southampton framework, to push this scale higher
across the charm threshold, and apply it to an exploration of the BK running
from 3 GeV to 9 GeV. This is done on Nf = 2+1+1 ensembles generated by the
RBC-UKQCD collaboration, and features a close study of the discretisation
effects.
