  A Two-Higgs doublet model (2HDM) can predict the observed muon $g-2$ for an
appropriately light pseudoscalar that now faces tight constraints. However, It
was shown in past that augmenting the 2HDM by an additional inert doublet can
lead to an explanation to the muon $g-2$ anomaly for a much heavier
pseudoscalar. In this study, we probe such a framework at the proposed
International Linear Collider (ILC) using beam polarization for $\sqrt{s}$ = 1
TeV. Using multivariate techniques, we analyse the signals $e^+ e^- \rightarrow
\tau^+ \tau^- +$ missing transverse energy and $e^+ e^- \rightarrow \mu^+ \mu^-
+ $ missing transverse energy in the lepton- and muon-specific versions of the
framework respectively. Our analysis reveals that the $e^+ e^-$ machine
operating at a 3000 fb$^{-1}$ luminosity predicts a 5$\sigma$ discovery of a
pseudoscalar as heavy as 400 GeV. Comparing with the previous study, it is
concluded that the ILC is a much more potent machine than the LHC in this
regard.
