Numerical characterization of some toric fiber bundles

Given a complex projective manifold X and a divisor D with normal crossings, we say that the logarithmic tangent bundle T_X(-log D) is R-flat if its pull-back to the normalization of any rational curve contained in X is the trivial vector bundle. If moreover -(K_X+D) is nef, then the log canonical divisor K_X+D is torsion and the maximally rationally chain connected fibration turns out to be a smooth locally trivial fibration with typical fiber F being a toric variety with boundary divisor D_|F .