Let p:X \to Y  be a map of smooth algebraic varieties.
Let C \subset T^*(X)   be a (locally closed) submanifold. Denote by p_*(C) \subset T^*(Y) the following set: 

$$\{(y,v) \in T^*(Y)|\exists x \in p^{-1}(y) \text{ with } (x,(d_x(p))^*(v)) \in C \}.$$

This opreration can describe (to some extent) what happens to the singular support of a D-module when one takes its direct image. 
 
My question is: when can one claim that $p_*$ of a (conic) Lagrangian manifold is Lagrangian? I heard it is not true in general. Is it true when p is proper? What are the counter examples?
 
 
Thank you very much,
Rami