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We construct a family of fibered threefolds $X_m \to (S , \Delta)$ such that$X_m$ has no \'etale cover that dominates a variety of general type but itdominates the orbifold $(S,\Delta)$ of general type. Following Campana, thethreefolds $X_m$ are called \emph{weakly special} but not \emph{special}. TheWeak Specialness Conjecture predicts that a weakly special variety defined overa number field has a potentially dense set of rational points. We prove that if$m$ is big enough the threefolds $X_m$ present behaviours that contradict thefunction field and analytic analogue of the Weak Specialness Conjecture. Weprove our results by adapting the recent method of Ru and Vojta. We alsoformulate some generalizations of known conjectures on exceptional loci thatfit into Campana's program and prove some cases over function fields.

Nonspecial varieties and Generalized Lang-Vojta conjectures