  In light of the recent CMS analysis on lepton flavor violating (LFV) heavy
Higgs searches and updated bounds on various search channels involving neutral
and charged scalars, we provide the updated constraints on the Type-III
Two-Higgs-Doublet-Model (2HDM) with a $\tau-\mu$ LFV. In doing so, we first
extend the CMS analysis to cover the mass region below 200 GeV by recasting
their data. After obtaining the bounds on the heavy Higgs production in the
mass range between 130 GeV and 450 GeV, we analyze the parameter space of the
Type-III 2HDM with various scenarios of mass spectrum and heavy Higgs
production strengths. We found that in most scenarios, searching for the heavy
Higgs in the mass range lower than $2 m_W$ is very important in constraining
the parameter space of the Type-III 2HDM. Hence, we suggest for the future
analysis that the search window for the LFV heavy Higgs be extended to the
lower mass region.
