Let $X$ be a scheme of finite type over $Spec(\mathbb{C})$. Let $X_{an}$ denote the associated complex analytic space. After fixing an isomorphism $\overline{\mathbb{Q}}_l\cong \mathbb{C}$, by $\S$6.1.2. of BBD (Asterisque 100) we know that there is fully faithfull functor

$$\mathcal{F}: D_c^b(X, \overline{\mathbb{Q}}_l) \rightarrow D_c^b(X_{an}, \mathbb{C}),$$

where the left hand side is the usual "derived category" of $l$-adic sheaves and the the right hand side is the usual derived category of complex sheaves with constructible (for an algebraic stratification) cohomology. As is well known this functor is not essentially surjective. However, on both sides we have two natural subcategories

$$Perv_{l}(X) \subset D_c^b(X, \overline{\mathbb{Q}}_l)$$

and

$$Perv_{\mathbb{C}}(X) \subset D_c^b(X_{an}, \mathbb{C}) $$ of l-adic Perverse sheaves and complex Perverse sheaves respectively. My question is this:

Does $\mathcal{F}$ induce an equivalence of categories between $Perv_{l}(X)$ and $Perv_{\mathbb{C}}(X)$?