Geometric Lang's conjecture

Let $X$ be a smooth projective variety of general type. Consider the irreducible positive-dimensional subvarieties of $X$ that are not of general type. Geometric Lang's conjecture. The union of all such subvarieties is a proper closed algebraic subset of $X$. This predicts that a general point of a variety of general type does not lie on any positive-dimensional subvariety failing to be of general type; the status of the assertion is not established by the supplied source context.