  While searching at the Large Hadron Collider (LHC) for the production and
decay of the CP-odd scalar ($A^0$) in the 2-Higgs-Doublet Model (2HDM) with
Natural Flavour Conservation (NFC) via the channels $gg\to A^0$ (through
one-loop triangle diagrams) and $A^0\to h^0 Z^*$ (with $m_{h^0} =125$ GeV or
$m_{h^0} < 125$ GeV, with $Z$ off-shell), respectively, a factorisation of the
two processes is normally performed, with the $A^0$ state being on-shell. While
this approach is gauge-invariant, it is not capturing the presence of either of
the following two channels: $gg\to Z^*\to h^0Z^*$ (through one-loop triangle
diagrams) or $gg\to h^0Z^*$ (through one-loop box diagrams). As the resolution
of the $A^0$ mass cannot be infinitely precise, we affirm that all such
contributions should be computed simultaneously, whichever the $h^0$($Z^{*}$)
decay(splitting) products, thereby including all possible interferences amongst
themselves. The cross section of the ensuing complete process is significantly
different from that obtained in the factorisation case, being of the order up
to ten percent in either direction at the integrated level and larger
(including changes in the shape of kinematical observables) at the differential
level. We thus suggest that the complete calculation ought to be performed
while searching for $A^0$ in this channel. We illustrate this need for the case
of a 2HDM of Type-I in the inverted hierarchy scenario with $m_{h^0}<125$ GeV.
