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We study the mechanism of orbital-order melting observed at temperature T_OOin the series of rare-earth manganites. We find that many-body super-exchangeyields a transition-temperature T_KK that decreases with decreasing rare-earthradius, and increases with pressure, opposite to the experimental T_OO. We showthat the tetragonal crystal-field splitting reduces T_KK further increasing thediscrepancies 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.

Orbital-order melting in rare-earth manganites: the role of super-exchange