  The {\em Planck} DR3 measurements of the temperature and polarization
anisotropies power spectra of the cosmic microwave background (CMB) show an
excess of smoothing of the acoustic peaks with respect to $\Lambda$CDM, often
quantified by a phenomenological parameter $A_{\rm L}$. A specific feature
superimposed to the primordial power spectrum has been suggested as a physical
solution for this smoothing excess. Here, we investigate the impact of this
specific localized oscillation with a frequency linear in the wavenumber,
designed to mimic the smoothing of CMB temperature spectrum corresponding to
$A_{\rm L} \simeq 1.1-1.2$ on the matter power spectrum. We verify the goodness
of the predictions in perturbation theory at next-to-leading order with a set
of N-body simulations, a necessary step to study the non-linear damping of
these primordial oscillations. We show that for a large portion of the
parameter space, the amplitude of this primordial oscillation can be strongly
damped on the observed nonlinear matter power spectrum at $z=0$, but a larger
signal is still persistent at $z \lesssim 2$ and is therefore a target for
future galaxy surveys at high redshifts. From an analysis of the BOSS DR12
two-point correlation function, we find ${\cal A}_{\rm lin} < 0.26$ at 95\% CL
by keeping the frequency fixed to the best-fit of {\em Planck} data.
