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In this article we present an updated spectrum-shape analysis of the $^{113}$Cd fourfold forbidden non-unique $β$-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 $β$-decays and adds to the indications for the existence of a quenching of $g_{\rm A}$ in low-momentum exchange nuclear processes.