  We study how large the rate of the lepton-flavor violating Higgs decay h->tau
mu can be in the (R-parity conserving) MSSM. We make no assumptions, such as
universality or alignment, about the flavor structure of the MSSM. We only
assume that all couplings and, in particular, the trilinear scalar ones, are
perturbative. We take into account lower bounds on the bino and slepton masses
from tau->mu gamma and h->gamma gamma as well as upper bounds on the trilinear
scalar couplings from the requirement that the global minimum is not charge
breaking. We find that in highly fine-tuned regions of parameter space, the
ratio BR(h->tau mu)/BR(h->tau tau) can be enhanced by about three orders of
magnitude above the estimate from naive dimensional analysis, but still about
two orders of magnitude below the present bound. Thus, if h->tau mu is
experimentally established to be close to present bounds, the MSSM will be
excluded.
