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 rational representation of $G$ (I guess it means a representation over $\mathbb{Q}$), satisfied some good conditions. Chapter II 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 $H^n_{dR}(Sh(G,X),\mathcal{V}(\xi),\nabla)$?

More precisely, these objects have canonical models over some suitable number field, say $L$. I want to study the $Gal(\bar{L}_v/L_v)$ representation $H^n_{et}(Sh(G,X)\times_L\bar{L}_v,V_\ell)$, for a place $v$ of $L$ above $\ell$. Moreover, the variation of Hodge structures should be considered, which will induce useful filtrations. I want to compare it with the algebraic de Rham cohomology $H^n_{dR}(Sh(G,X)/L,\mathcal{V}(\xi),\nabla)$

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