  We analyse new signals of a 3-Higgs Doublet Model (3HDM) at the Large Hadron
Collider (LHC) where only one doublet acquires a Vacuum Expectation Value
(VEV), preserving a $Z_2$ parity. The other two doublets are \textit{inert} and
do not develop a VEV, leading to a \textit{dark scalar sector} controlled by
$Z_2$, with the lightest CP-even dark scalar $H_1$ being the Dark Matter (DM)
candidate. This leads to the loop induced decay of the next-to-lightest scalar,
$H_2 \to H_1 \ell \bar \ell $ ($\ell =e,\mu$), mediated by both dark CP-odd
neutral and charged scalars. This is a smoking-gun signal of the 3HDM since it
is not allowed in the 2-Higgs Doublet Model (2HDM) with one inert doublet and
is expected to be important when $H_2$ and $H_1$ are close in mass. In
practice, this signature can be observed in the cascade decay of the SM-like
Higgs boson, $h\to H_1 H_2\to H_1 H_1 \ell \bar \ell$ into two DM particles and
di-leptons or $h\to H_2 H_2\to H_1 H_1 \ell \bar \ell \ell \bar \ell$ into two
DM particles and four-leptons, where $h$ is produced from gluon-gluon Fusion.
In order to test the feasibility of these channels at the LHC, we devise some
benchmarks, compliant with collider, DM and cosmological data, for which the
interplay between these production and decay modes is discussed. In particular,
we show that the resulting detector signatures, $\Et \ell \bar \ell$ or $\Et
\ell \bar \ell \ell \bar \ell$, with the invariant mass of $ \ell \bar \ell$
pairs much smaller than $m_Z$, can potentially be extracted already from Run 3
data and at the High-Luminosity phase of the LHC.
