  In this article we present an updated spectrum-shape analysis of the
$^{113}$Cd fourfold forbidden non-unique $\beta$-decay transition in order to
address the quenching of the weak axial-vector coupling $g_{\rm A}$ in
low-momentum exchange nuclear processes. The experimental data were collected
in a dedicated low-threshold run with the COBRA demonstrator at the LNGS and
resulted in 44 individual $^{113}$Cd spectra. These data are evaluated in the
context of three nuclear model frameworks based on a revised version of the
spectrum-shape method and the conserved vector current hypothesis. The novel
idea devised in the present work is to fit the value of the small relativistic
nuclear matrix element (s-NME) driving the nuclear model calculations, which
remained essentially as a free parameter in previous studies. This is done by
tuning the nuclear structure calculations and making use of the interplay of
$g_{\rm A}$ and the s-NME such that the experimentally known $^{113}$Cd
half-life gets reproducible by the different frameworks. In this way, a best
fit s-NME value can be derived for each of the considered nuclear models, which
finally enters the template calculations used to perform the spectrum-shape
analysis for each of the obtained $^{113}$Cd spectra. The primary analysis
strategy results in significantly quenched values of the axial-vector coupling
for all three nuclear models: $\overline{g}_{\rm A}(\text{ISM}) = 0.907 \pm
0.064$, $\overline{g}_{\rm A}(\text{MQPM}) = 0.993 \pm 0.063$ and
$\overline{g}_{\rm A}(\text{IBFM-2}) = 0.828 \pm 0.140$. Moreover, with our
data-driven approach one of the main shortcomings of the spectrum-shape method
has been resolved. This achievement is a milestone in the description of
strongly forbidden $\beta$-decays and adds to the indications for the existence
of a quenching of $g_{\rm A}$ in low-momentum exchange nuclear processes.
