The $\eta^\prime$-meson energy spectrum in the inclusive $\Upsilon (1S) \to \eta^{(\prime)} g g g \to \eta^{(\prime)} + X$ decay measured by the CLEO Collaboration in 2002 allowed one to constrain the lowest Gegenbauer coefficients $B^q_2$ and $B^g_2$ of the quark-antiquark and gluonic distribution amplitudes of the $\eta^\prime$-meson entering the $\eta^\prime g^* g$ effective vertex function (EVF). The fitting procedure of the CLEO data on the hard part of the $\eta^\prime$-meson energy spectrum was based on the theoretical expression calculated in the leading-order perturbative QCD in the static-quark limit for the orthoquarkonium. The resulting constraints were combined with the existing ones on these coefficients from an analysis of the $\eta^\prime - \gamma$ transition form factor. The updated measurements of the $\eta^\prime$-meson energy spectrum by the CLEO Collaboration in 2006 are not in accord with the so-calculated theoretical $\eta^\prime$-energy spectrum. As a first step, we incorporate the dependence of the strong coupling constant in the $\eta^\prime g^* g$ EVF on the quark energy and determine the Gegenbauer coefficients. With this values we compute the $\eta$- and $\eta$-meson energy spectra in the inclusive decay $\Upsilon (1S) \to \eta^{(\prime)} g g g \to \eta^{(\prime)} + X$, which can be tested against the high statistics Belle and Belle-II data from $\Upsilon (1S)$-decays.