# Singular support of an irreducible perverse sheaf

I was studying Sheaves on Manifolds by Kashiwara and Schapira, and while the singular support seems like a complicated invariant I cannot seem to find a counterexample to the following:

Let $$X$$ be a smooth complex variety and $$\mathcal{F}=IC(U,\mathcal{L})$$ be an irreducible perverse sheaf, where $$\mathcal{L}$$ is a local system on $$U\subset X$$. Then $$SS(\mathcal{F})=T_{\overline{U}}^*X$$, where the latter means the conormal bundle at $$\overline{U}$$.

This seems too easy of an answer to be true, but I still cannot find either a counterexample or a proof, and I cannot think of how to get an explicit answer using the Riemann-Hilbert correspondance either. Any help?

PS. I have asked the question already in Stack Exchange but it wasn't answered and I thought it may be more appropriate here after all?

Take $$X=\mathbb C$$, $$U=\mathbb C^\times$$, and $$\mathcal L$$ a nontrivial rank 1 local system (with monodromy $$\mu \neq 1$$, say).
Then the singular support of $$IC(U,\mathcal L)$$ is the union $$T^\ast _X X \cup T^\ast_0 X$$ of the zero section (which is what your conjecture would give in this case) with the cotangent fiber.
Note that here we have $$j_\ast \mathcal L \cong IC(U,\mathcal L) \cong j_! \mathcal L$$. One can compute the singular support either using the sheaf definition from Kashiwwara-Schapira, or by considering the associated $$D$$-module $$D_{\mathbb C}/D_{\mathbb C}(x\partial_x - \log(\mu))$$ under the Riemann-Hilbert correspondence.
My impression is that the singular support of a perverse sheaf is the characteristic variety of the regular holonomic $$D$$-module corresponding to it under Riemann-Hilbert. Assuming that's the case, it is possible to answer this in the negative. Let $$X$$ be the disk or the affine line if you prefer, and $$U=X-\{0\}$$. Choose a nontrivial rank one local system $$\mathcal{L}$$ on $$U$$. By Riemann-Hilbert, there is regular connection $$\nabla$$ on $$\mathcal{O}_X$$ such that $$\mathcal{L}=\ker\nabla$$ over $$U$$. Then $$IC(U,L)= j_*L$$ (or a translate depending on your convention), and the corresponding $$D$$-module $$M$$ is the minimal extension of $$\nabla$$. The characteristic variety of $$M|_U$$ is the zero section of $$T_U^*$$. Therefore the characteristc variety of $$M$$ is either the zero section of $$T^*_X$$ or the zero section union $$T_0^*$$. If it's the former, then $$M$$ would have to be a connection which contradicts our choice, so it must be the latter.