  We study the mechanism of orbital-order melting observed at temperature T_OO
in the series of rare-earth manganites. We find that many-body super-exchange
yields a transition-temperature T_KK that decreases with decreasing rare-earth
radius, and increases with pressure, opposite to the experimental T_OO. We show
that the tetragonal crystal-field splitting reduces T_KK further increasing the
discrepancies with experiments. This proves that super-exchange effects,
although very efficient, in the light of the experimentally observed trends,
play a minor role for the melting of orbital ordering in rare-earth manganites.
