  We consider a restricted Type-I seesaw scenario with four texture zeros in
the neutrino Yukawa matrix, in the weak basis where both the charged-lepton
Yukawa matrix and the Majorana mass matrix for right-handed neutrinos are
diagonal and real. Inspired by grand unified theories, we further require the
neutrino Yukawa matrix to exhibit a similar hierarchical pattern to that in the
up-type quark Yukawa matrix. With such a hierarchy requirement, we find that
leptogenesis, which would operate in a $N_2^{}$-dominated scenario with the
asymmetry generated by the next-to-lightest right-handed neutrino $N_2^{}$, can
greatly reduce the number of allowed textures, and disfavors the scenario that
three light neutrinos are quasi-degenerate. Such a quasi-degenerate scenario of
light neutrinos may soon be tested in upcoming neutrino experiments.
