  Unlike other realisations of the 2-Higgs Doublet Model (2HDM), the so-called
Type-I allows for a very light Higgs boson spectrum. Specifically, herein, the
heaviest of the two CP-even neutral Higgs states, $H$, can be the one
discovered at the Large Hadron Collider (LHC) in 2012, with a mass of $\approx
125$ GeV and couplings consistent with those predicted by the Standard Model
(SM). In such a condition of the model, referred to as `inverted mass
hierarchy', the decay of the SM-like Higgs state into pairs of the lightest
CP-even neutral Higgs boson, $h$, is possible, for masses of the latter ranging
from $M_H/2\approx 65$ GeV down to 15 GeV or so, all compatible with
experimental constraints. In this paper, we investigate the scope of the LHC in
accessing the process $gg\to H \to hh\to b\bar b\tau\tau$ by performing a Monte
Carlo (MC) analysis aimed at extracting this signal from the SM backgrounds, in
presence of a dedicated trigger choice and kinematic selection. We prove that
some sensitivity to such a channel exists already at Run 3 of the LHC while the
High-Luminosity LHC (HL-LHC) will be able to either confirm or disprove this
theoretical scenario over sizable regions of its parameter space.
