On a conjecture of Deligne

Let $X$ be a smooth variety over $\mathbb{F}_p$. Let $E$ be a number field. For each nonarchimedean place $\lambda$ of $E$ prime to $p$ consider the set of isomorphism classes of irreducible lisse $\overline{E}_{\lambda}$-sheaves on $X$ with determinant of finite order such that for every closed point $x\in X$ the characteristic polynomial of the Frobenius $F_x$ has coefficents in $E$. We prove that this set does not depend on $\lambda$.
The idea is to use a method developed by G. Wiesend to reduce the problem to the case where $X$ is a curve. This case was treated by L. Lafforgue.