  We study the Large Hadron Collider (LHC) search potential of a
$\mathbb{Z}_4$-based two Higgs doublet model which can simultaneously explain
the muon $g-2$ anomaly and the observed dark matter. The neutral scalars in the
second Higgs doublet couple to $\mu$ and $\tau$ and largely contribute to the
muon anomalous magnetic moment through the one-loop diagram involving $\tau$
and scalars. An additional singlet scalar which is charged under the discrete
symmetry can be a dark matter candidate. An upper limit on the scalar mass
originates from the unitarity constraint, and the $\mu\tau$ flavor-violating
nature of the scalars predicts non-standard signatures at the LHC. However, the
previously proposed $\mu^\pm\mu^\pm\tau^\mp\tau^\mp$ signal via the electroweak
heavy neutral scalar pair production at the LHC loses sensitivity for
increasing scalar mass. We revisit this model and investigate the LHC prospects
for the single production of the $\mu\tau$ flavor-violating neutral scalar. It
is shown that the single scalar process helps to extend the LHC reach to the
1$\,$TeV mass regime of the scenario. The search potential at the high energy
LHC is also discussed.
