  The type-I two-Higgs-doublet model in the inverted Higgs scenario can retain
the theoretical stability all the way up to the Planck scale. The Planck-cutoff
scale, $\Lambda_{\rm cut}^{\rm Planck}$, directly impacts the mass spectra such
that all the extra Higgs boson masses should be light below about 160 GeV.
However, the observation of the light masses of new Higgs bosons does not
indicate the high cutoff scale because a low cutoff scale can also accommodate
the light masses. Over the viable parameter points that satisfy the theoretical
requirements and the experimental constraints, we show that the trilinear Higgs
couplings for low $\Lambda_{\rm cut}$ are entirely different from those for the
Planck-cutoff scale. The most sensitive coupling to the cutoff scale is from
the $h$-$h$-$h$ vertex, where $h$ is the lighter CP-even Higgs boson at a mass
below 125 GeV. Among the multi-Higgs productions mediated by Higgs bosons, the
gluon fusion processes of $gg \to h h $ and $gg \to AA$ are insensitive to the
cutoff scale, yielding a small variation of $\mathcal{O}(1)\,{\rm fb}$
according to $\Lambda_{\rm cut}$. The smoking-gun signature is from the triple
Higgs production of $q\bar{q}' \to W^* \to H^\pm hh$, which solely depends on
the $h$-$h$-$h$ vertex. The cross section for $\Lambda_{\rm cut}=1\,{\rm TeV}$
is about $10^3$ times larger than that for the Planck-cutoff scale. Since the
decay modes of $H^\pm \to W^* h/W^* A$ and $h/A \to bb$ are dominant, the
process yields the $6b+\ell\nu$ final state, which enjoys an almost
background-free environment. Consequently, the precision measurement of $pp \to
H^\pm hh$ can probe the cutoff scale of the model.
