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?