  The usual description of self-induced neutrino flavor conversions in core
collapse supernovae (SNe) is based on the dominance of the neutrino density
n_nu over the net electron density n_e. However, this condition is not met
during the post-bounce accretion phase, when the dense matter in a SN is piled
up above the neutrinosphere. As recently pointed-out, a dominant matter term in
the anisotropic SN environment would dephase the flavor evolution for neutrinos
traveling on different trajectories, challenging the occurrence of the
collective behavior in the dense neutrino gas. Using the results from recent
long term simulations of core-collapse SN explosions, based on three flavor
Boltzmann neutrino transport in spherical symmetry, we find that both the
situations of complete matter suppression (when n_e >> n_nu) and matter-induced
decoherence (when n_e \gtrsim n_nu) of flavor conversions are realized during
the accretion phase. The matter suppression at high densities prevents any
possible impact of the neutrino oscillations on the neutrino heating and hence
on the dynamics of the explosion. Furthermore, it changes the interpretation of
the Earth matter effect on the SN neutrino signal during the accretion phase,
allowing the possibility of the neutrino mass hierarchy discrimination at not
too small values of the leptonic mixing angle \theta_{13} (i.e.
\sin^2{\theta}_{13} \gtrsim 10^{-3}).
