  The lepton-sepcific (or type X) 2HDM (L2HDM) is an attractive new physics
candidate explaining the muon $g-2$ anomaly requiring a light CP-odd boson $A$
and large $\tan\beta$. This scenario leads to $\tau$-rich signatures, such as
$3\tau$, $4\tau$ and $4\tau+W/Z$, which can be readily accessible at the LHC.
We first study the whole L2HDM parameter space to identify allowed regions of
extra Higgs boson masses as well as two couplings $\lambda_{hAA}$ and $\xi_h^l$
which determine the 125 GeV Higgs boson decays $h\to \tau^+\tau^-$ and $h\to
AA/AA^*(\tau^+\tau^-)$, respectively. This motivates us to set up two regions
of interest: (A) $m_A \ll m_{H} \sim m_{H^\pm}$, and (B) $m_A \sim m_{H^\pm}
\sim {\cal O}(100) \mbox{GeV} \ll m_H$, for which derive the current
constraints by adopting the chargino-neutralino search at the LHC8, and then
analyze the LHC14 prospects by implementing $\tau$-tagging algorithm. A
correlated study of the upcoming precision determination of the 125 GeV Higgs
boson decay properties as well as the observation of multi-tau events at the
next runs of LHC will be able to shed light on the L2HDM option for the muon
$g-2$.
