  During the accretion phase of a core-collapse supernova (SN), the
deleptonization flux has recently been found to develop a global dipole pattern
(LESA---Lepton Emission Self-sustained Asymmetry). The $\nu_e$ minus
$\bar\nu_e$ flux essentially vanishes in one direction, potentially
facilitating self-induced flavor conversion. On the other hand, below the
stalled shock wave, self-induced flavor conversion is typically suppressed by
multi-angle matter effects, preventing any impact of flavor conversion on SN
explosion dynamics. In a schematic model of SN neutrino fluxes, we study the
impact of modified $\bar\nu_e$-$\nu_e$ flux asymmetries on collective flavor
conversion. In the parameter space consisting of matter density and effective
neutrino density, the region of instability with regard to self-induced flavor
conversion is much larger for a vanishing lepton number flux, yet this
modification does not intersect a realistic SN profile. Therefore, it appears
that, even in the presence of LESA, self-induced flavor conversion remains
suppressed below the shock front.
