Take $X\to V$ a closed embedding, where $X$ is not necessarily smooth, $V$ is affine and smooth. Define the category $\mathcal{C}$ of $\mathcal{D}$ modules on $X$ to be the full subcategory of $\mathcal{D}$ modules on $V$ with support on $X$.
I want to ask if the inclusion $D(\mathcal{C})\to D_\mathcal{C}(\mathcal{D}_V\text{-mod})$ is an equivalence (where $D_\mathcal{C}(\mathcal{D}_V\text{-mod})$ is the full subcategory of $D(\mathcal{D}_V\text{-mod})$ consisting complexes with cohomologies in $\mathcal{C}$). I believe it is. In fact for what I really need, I only need the inclusion to be fully faithful, but nevertheless it should be an equivalence. The derived category is the one with either quasi-coherent or coherent objects, and should be bounded. Note when $X$ is smooth, this is just a version of Kashiwara's Theorem.
What we know: $\mathcal{C}$ is thick/Serre, $\mathcal{C}$ and $\mathcal{D}_V\text{-mod}$ are Grothedieck (so has enough injectives).
There is a potential useful theorem in Kashiwara-Schapira Category and Sheaves, Theorem 13.2.8, but I don't know how to show the conditions. Also, these overflow questions can be useful: Equivalence between a derived subcategory and a subcategory of the derived category