  This paper presents a study of a possible contribution to a Higgs boson
signal in the $hh\to \gamma\gamma\gamma\gamma$ channel due to $H\to hh$ decays,
in the framework of the CP-conserving 2-Higgs Doublet Model Type-I (2HDM-I),
where the heavier of the two CP-even Higgs bosons defined herein, $H$, is the
SM-like Higgs state observed with a mass of 125 GeV at the Large Hadron
Collider (LHC). We perform a broad scan of the 2HDM-I parameter space, in
presence of both up-to-date theoretical and experimental constraints, in order
to extract the interesting regions yielding such a signal. Then, after
validating our numerical framework against public experimental analyses carried
out at the LHC, we proceed to assess its scope in constraining and/or
extracting the $gg\to H\to hh\to \gamma\gamma\gamma\gamma$ signal in presence
of a sophisticated Monte Carlo (MC) simulation. We find that, over a
substantial region of the 2HDM-I parameter space presently un-accessible, the
LHC will be able to establish such a potential signature in the next 2--3
years.
