  We show that standard matter effects in the outer layers of core-collapse
supernovae significantly constrain the flavor composition of the neutrino flux,
even with the enormous uncertainties originating from self-induced flavor
conversions in the supernova core. Under certain conditions, the neutrino flux
resulting from self-induced conversions can be represented as a combination of
flavor eigenstates in an arbitrary flavor ratio configuration. In this
scenario, we find that, for the normal mass ordering, the fraction of neutrinos
with electron flavor reaching the Earth, denoted as $f_{\nu_e}$, is constrained
to be less than $0.5$ for all energies throughout the emission phase, whereas,
for inverted mass ordering, we anticipate neutrinos arriving in near flavor
equipartition ($f_{\nu_e}\approx 1/3$). In case adiabaticity is violated in the
region of standard matter effects, the result is flavor equipartition for both
mass orderings. Subsequently, we elaborate on the impact of wave-packet
decoherence during self-induced conversions and explore alternative scenarios
that could affect the aforementioned results.
