I've got a negative answer from Thomas Bitoun:

<cite authors="Malgrange, Bernard">_Malgrange, Bernard_, [**Sur les images directes de ${\mathcal D}$-modules**](https://doi.org/10.1007/BF01168827), Manuscr. Math. 50, 49–71 (1985). [Zbl&nbsp;0572.32014](https://zbmath.org/0572.32014).</cite>

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.