  We calculated in nonlinear bosonized theory $1/\bar{N}$ corrections to the
Landau Fermi-liquid fixed-point (FLFP) axial-vector coupling constant in
nuclear matter $g_A^L\approx 1$ to which the Landau parameter $F_1^\omega$
predominantly contributes. We obtain the correction to $F_1^\omega$ to
calculate the correction $\delta g_A^L$ to the axial-vector coupling constant
$g_A^L$ at the nuclear saturation density. It comes out to be extremely small,
$\delta g_A^L\sim O(10^{-4})$. We discuss how the "dilaton-limit fixed-point
(DLFP)" result $g_A=1$ can be preserved from finite nuclei to high densities
relevant to massive neutron stars and its possible impact on $0\nu\beta\beta$
decay processes involved in going beyond the Standard Model.
