Let $X$ a smooth variety over a field of caracteristic zero. Let $M$ be in the derived category of holonomic $\mathcal{D}_{X}$-modules, $D^{b}_{h}(\mathcal{D}_{X})$.

We know that if we assume that $M$ is regular then the map given by the de Rahm functor:

$DR_{X}(M):RHom_{\mathcal{D}_{X}}(M,M)\rightarrow RHom_{\mathbb{C}_{X}}(DR(M),DR(M))$

It's proved in the corollary 3.1.15 at the IHES paper of Mebkhout.

Do we have the converse? Say, if $DR_{X}(M)$ is an isomorphism, then M is regular?

Let $X$ be a smooth variety over a field of characteristic zero. Let $M$ be in the derived category of holonomic $\mathcal{D}_{X}$-modules, $D^{b}_{h}(\mathcal{D}_{X})$.

We know that if we assume that $M$ is regular then the map given by the de Rham functor,

$DR_{X}(M):RHom_{\mathcal{D}_{X}}(M,M)\rightarrow RHom_{\mathbb{C}_{X}}(DR(M),DR(M)),$

is an isomorphism. It's proved in the corollary 3.1.15 at the IHES paper of Mebkhout.

Do we have the converse? Say, if $DR_{X}(M)$ is an isomorphism, then M is regular?