  For exponents $p$ satisfying $0<|p-3|\ll 1$ and only in the context of
spatially even solutions 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. The proof is similar to a related result of
Martel, preprint arXiv:2312.11016, for a cubic quintic NLS. Here we modify the
second part of Martel's argument, replacing the second virial inequality for a
transformed problem with a smoothing estimate on the initial problem,
appropriately tamed by multiplying the initial variables and equations by a
cutoff.
