Fix a smooth complex algebraic variety $X$, and $\mathcal D_X$ its sheaf of differential operators.
If a $\mathcal D_X$-module $\mathcal M$ is holonomic, it has finite length. The usual proof uses the total multiplicity of the characteristic variety $\text{Ch} (\mathcal M)$ : if the length of $\mathcal M$ is greater than $1$, you can find a strict submodule $\mathcal N$, which is coherent because $\mathcal D_X$ is Noetherian, and thus holonomic. $\mathcal N$ has a smaller total multiplicity because $\mathcal M / \mathcal N$ is nonzero, induction, QED.
This same proof, however, fails if $X$ is a complex analytic manifold : $\mathcal D_X$ fails to be noetherian, being merely $\mathcal D_X$-coherent as a left (and right) $\mathcal D_X$-module. Therefore, a coherent $\mathcal D_X$-module may a priori have no coherent submodules but many quasicoherent ones.
My question is the following : does the statement still hold as it is, or do you need to define the coherent length of a coherent $\mathcal D_X$-module (i.e. composition series consisting of coherent submodules) ? In that last case, the proof from the algebraic setting works fine.
If we're not using that second definition, something like this sounds like it could work : you can locally take a section of $\mathcal M$ that does not generate the whole thing (assuming $\mathcal M$ has length $> 1$), and you have a local submodule of $\mathcal M$ that is $\mathcal D_X$-coherent because it is a finitely generated submodule of a coherent module. The question then lies in knowing whether or not one can extend a coherent $\mathcal D_U$-module into a coherent $\mathcal D_X$-module. I know it is possible in a wide array of cases for quasicoherent and coherent $\mathcal O_X$-modules (for instance, here are scheme-theoretic versions of such theorems), but I am not aware of similar results for analytic $\mathcal D_X$-modules.
The question might not even be relevant because from what I've gathered, quasi-coherent ($\mathcal O_X$ or $\mathcal D_X$) modules are ill-defined in the analytic world, so I guess that's also part of the question : does there even exist a relevant definition of length that isn't "composition series by coherent submodules" ?
I've looked at many courses and reviews, and the definition of length of a coherent $\mathcal D_X$-module is always skipped or given just in the algebraic case.
