  In this article we study the full one-loop SUSY contributions to the lepton
flavor violating Higgs decay $h \to \tau \bar \mu $, within the context of the
supersymmetric inverse sesaw model. We assume that both the right-handed
neutrino masses, $M_R$, and their supersymmetric partner masses, $m_{\tilde
\nu_R}$, are not far from the interesting ${\cal O}({\rm TeV})$ energy scale,
and we work with scenarios with large neutrino Yukawa couplings that transmit
large lepton flavor violating effects. By exploring the behavior with the most
relevant parameters, mainly $M_R$, $m_{\tilde \nu_R}$ and the trilinear
sneutrino coupling $A_\nu$, we will look for regions of the parameter space
where the enhancement of $\mathrm{BR}(h\rightarrow \tau \bar \mu)$ is large
enough to reach values at the percent level, which could explain the excess
recently reported by CMS and ATLAS at the CERN Large Hadron Collider.
