Abstract: Neutrinos propagating in dense neutrino media such as core-collapse supernovae and neutron star merger remnants can experience the so-called fast flavor conversions on scales much shorter than those expected in vacuum. A very generic class of fast flavor instabilities is the ones which are produced by the backward scattering of neutrinos off the nuclei at relatively large distances from the supernova core. In this study we demonstrate that despite their ubiquity, such fast instabilities are unlikely to cause significant flavor conversions if the population of neutrinos in the backward direction is not large enough. Indeed, the scattering-induced instabilities can mostly impact the neutrinos traveling in the backward direction, which represent only a small fraction of neutrinos at large radii. We show that this can be explained by the shape of the unstable flavor eigenstates, which can be extremely peaked at the backward angles.