This is more a remark than an answer.


It is perhaps worth noting that in the related classification of foliations on surfaces by McQuillan, Brunella, and Mendes abundance does not hold. The so called Hilbert Modular foliations are examples of foliations with nef canonical bundle but with Kodaira-Iitaka
dimension negative. These turn out to be the only examples, and the proof of this fact is the harder part of the classification.