Large BR(h->tau mu) in the MSSM?

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.