Rational connectedness modulo the Non-nef locus

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It is well known that a smooth projective Fano variety is rationally connected. Recently Zhang (and later Hacon and McKernan as a special case of their work on the Shokurov RC-conjecture) proved that the same conclusion holds for a klt pair (X,) such that -(Kₓ+) is big and nef. We prove here a natural generalization of the above result by dropping the nefness assumption. Namely we show that a klt pair (X,) such that -(Kₓ+) is big is rationally connected modulo the non-nef locus of -(Kₓ+). This result is a consequence of a more general structure theorem for arbitrary pairs (X,) with -(Kₓ+) pseff.