  In a previous work we developed a search strategy for staus produced by the
decay of the heavy CP-even Higgs boson $H$ within the context of the large
$\tan\beta$ regime of the minimal supersymmetric standard model (MSSM) in an
scenario of large stau mixing. Here we study the performance of such search
strategy by confronting it with the complementary mixing pattern in which
decays of both the CP-even and CP-odd heavy Higgs bosons contribute to the
production of $\widetilde{\tau}_1\widetilde{\tau}_2^{*} \;+\; c.c$ pairs.
Again, we focus on final states with two opposite-sign tau leptons and large
missing transverse energy. We find that our proposed search strategy, although
optimized for the large stau mixing scenario, is still quite sensitive to the
complementary mixing pattern. For instance, with a total integrated luminosity
of only 100 fb$^{-1}$ we are able to exclude heavy Higgs masses above 850 GeV
for average stau masses higher than 290 GeV. We also extend the results
reported in the preceding work for the large mixing scenario by including now
the exclusion limits at 100 fb$^{-1}$ and the prospects both for exclusion and
discovery in a potential high luminosity phase of the LHC (1000 fb$^{-1}$).
Finally, we discuss the possibility to distinguish the two mixing scenarios
when they share the same relevant mass spectrum and both reach the discovery
level with our search strategy.
