Numerical characterization of some toric fiber bundles

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