  Assuming the nonlinear Fermi Golden Rule (FGR) and no resonance at the
threshold of the continuous spectrum of the linearization and assuming
furthermore as hypotheses the results proved numerically by Chang et al.
\cite{Chang} for the exponent $p\in (3,5)$, we prove that the ground states of
the nonlinear Schr\"odinger equation (NLS) with pure power nonlinearity of
exponent $p$ in the line are asymptotically stable for a certain set of values
of the exponent $p$ where the FGR occurs by means of a discrete mode 3rd or 4th
order power interaction with the continuous mode. The argument is similar to
our recent result valid for $p$ near 3 contained in \cite{CM24D1}.
