  When several four-quark operators are allowed to mix through renormalisation,
this can considerably amplify the problems coming from perturbative truncation
and discretisation effects. In this work we investigate whether our previous
$B_K$ strategy can conveniently be generalised to a wider set of operators,
corresponding to the so-called "SUSY $B_K$" basis of four-quark operators. We
show that the discretisation effects, when plotted as a function of $ap$, are
surprisingly independent of the lattice spacing. They appear reasonably under
control up to very large energy scales. This allows us to discuss the effect of
varying the intermediate scale on which the perturbative matching is done, and
therefore the prospects of future high-precision studies with a
Rome-Southampton renormalisation.
