This question is based on Milne "canonical models of Shimura varieties and automorphic vector bundles"

Let $(G,X)$ be a Shimura datum, $(V,\xi)$ be a representation of $G$, satisfied some good conditions. Prop. 3.3 in Milne's paper says there are

(1) $\mathbb{Q}_\ell$-coefficient local system $V_\ell$ on $Sh(G,X)_{et}$, for each prime $\ell$.

(2) a vector bundle $\mathcal{V}(\xi)$ on $Sh(G,X)$, with a flat connection $\nabla$.

Is there any comparison theorem relating $H^n_{et}(Sh(G,X),V_\ell)$ with $H^n_{dR}(Sh(G,X),\mathcal{V}(\xi),\nabla)$?

For example, if $(V,\xi)$ is the trivial representation, and $\ell$ is "good", then we can apply Falting's comparison theorem.