  We present a reinterpretation study of existing results from the CMS
Collaboration, specifically, searches for light Beyond the Standard Model (BSM)
Higgs pairs produced in the chain decay $pp\to H_{\rm SM}\to hh(aa)$ into a
variety of final states, in the context of the CP-conserving 2-Higgs Doublet
Model (2HDM) Type-I. Through this, we test the Large Hadron Collider (LHC)
sensitivity to a possible new signature, $pp\to H_{\rm SM}\to ZA\to ZZ h$, with
$ZZ\to jj \mu^+\mu^-$ and $h\to b\bar b$. We perform a systematic scan over the
2HDM Type-I parameter space, by taking into account all available theoretical
and experimental constraints, in order to find a region with a potentially
visible signal. We investigate the significance of it through a full Monte
Carlo simulation down to the parametrised detector level. We show that such a
signal is an alternative promising channel to standard four-body searches for
light BSM Higgses at the LHC already with an integrated luminosity of ${\cal L}
= 300~{\rm fb}^{-1}$. For a tenfold increase of the latter, discovery should be
possible over most of the allowed parameter space.
