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.
