  Within the $\nu$MSSM, a Minimal Supersymmetric neutrino See-saw Model, Lepton
Flavour Violating Higgs couplings are strongly enhanced at large $\tan\beta$
($\gsim30$), which can lead to BR$(H^0/A^0 \to \tau\mu) \simeq O(10^{-4})$, for
$M_{H^0/A^0}\gsim 160$ GeV. Enhancements on the production of Higgs bosons,
through the gluon fusion mechanism, $gg\to H^0/A^0$, and the associated
production channel $gg,q\bar q\to b\bar bH^0/A^0$, whose rates grow with
$\tan\beta$, as well as the mass degeneracy that occurs between the $H^0$ and
$A^0$ states in this regime, also contribute to further the possibilities to
detect a heavy Higgs signal into $\tau\mu$ pairs. We show that the separation
of $\tau\mu$ Higgs events from the background at the upcoming CERN Large Hadron
Collider could be done for Higgs masses up to about 600 GeV for 300 fb$^{-1}$
of luminosity, for large $\tan\beta$ values. However, even with as little as 10
fb$^{-1}$ one can probe $H^0/A^0$ masses up to 400 GeV or so, if
$\tan\beta=60$. Altogether, these processes then provide a new Higgs discovery
mode as well as an independent test of flavour physics.
