  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 it
dominates the orbifold $(S,\Delta)$ of general type. Following Campana, the
threefolds $X_m$ are called \emph{weakly special} but not \emph{special}. The
Weak Specialness Conjecture predicts that a weakly special variety defined over
a 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 the
function field and analytic analogue of the Weak Specialness Conjecture. We
prove our results by adapting the recent method of Ru and Vojta. We also
formulate some generalizations of known conjectures on exceptional loci that
fit into Campana's program and prove some cases over function fields.
