Direct image of Lagrangian subspaces of the co-tangent bundle

Question

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:

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

This operation 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?

Answer 1

I've got a negative answer from Thomas Bitoun:

Malgrange, Bernard, Sur les images directes de ${\mathcal D}$-modules, Manuscr. Math. 50, 49–71 (1985). Zbl 0572.32014.

It is at the very end of the paper, (6.3) page 23 of the electronic version. It is in French and Malgrange attributes the example to Kashiwara.