  The two-dimensional supersymmetric $\s$-model on a K\"ahler manifold has a
non-vanishing $\b$-function at four loops, but the $\b$-function at five loops
can be made to vanish by a specific choice of renormalisation scheme. We
investigate whether this phenomenon persists at six loops, and conclude that it
does not; there is a non-vanishing six-loop $\b$-function irrespective of
renormalisation scheme ambiguities.
