  The two-Higgs-doublet model with a $U(1)_H$ gauge symmetry (N2HDM-$U(1)$) has
several advantages compared to the ``standard'' $Z_2$ version (N2HDM-$Z_2$): It
is purely based on gauge symmetries, involves only spontaneous symmetry
breaking, and is more predictive because it contains one parameter less in the
Higgs potential, which further ensures $CP$ conservation, i.e., avoiding the
stringent bounds from electric dipole moments. After pointing out that a
second, so far unknown version of the N2HDM-$U(1)$ exists, we examine the
phenomenological consequences for the Large Hadron Collider (LHC) of the
differences in the scalar potentials. In particular, we find that while the
N2HDM-$Z_2$ predicts suppressed branching ratios for decays into different
Higgs bosons for the case of the small scalar mixing (as suggested by Higgs
coupling measurements), both versions of the N2HDM-$U(1)$ allow for sizable
rates. This is particularly relevant in light of the CMS excess in resonant
Higgs-pair production at around $650\,$GeV of a Standard Model Higgs boson
subsequently decaying to photons and a new scalar with a mass of
$\approx90\,$GeV subsequently decaying to bottom quarks (i.e., compatible with
the CMS and ATLAS $\gamma\gamma$ excesses at $95\,$GeV and $\approx 670\,$GeV).
As we will show, this excess can be addressed within the N2HDM-$U(1)$ in case
of a nonminimal Yukawa sector, predicting an interesting and unavoidable $Z+
b\bar b$ signal and motivating further asymmetric di-Higgs searches at the LHC.
