  In this article we consider log canonical pairs which are log-smooth. If the
corresponding canonical bundle is pseudo-effective, then we show that any
quotient of the orbifold cotangent bundle of the pair has a pseudo-effective
determinant. One of the new ingredients in the proof is a generalization of the
Bogomolov-McQuillan algebraicity criterion in the context of holomorphic
foliations whose minimal slope with respect to a movable class is positive.
