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DoesDoesn't this paper give ana positive answer to your question? (And some more I think)
Qi Zhang, Rational connectedness of log $Q$-Fano varieties, J. Reine Angew. Math. 590 (2006)
Does this paper give an answer to your question? (And some more I think)
Doesn't this paper give a positive answer to your question? (And some more I think)
