Let $k$ be a finite field and $l$ be different from characteristic of $k$. Is there a realization functor from the Voevodsky's category with $\mathbb Q$ coefficients to the constructible mixed étale category of sheaves over $k$ such that $M_{gm}(X)\mapsto \mathcal Rp_*p^*1_k$ for *any* scheme $X$ (of finite type, seperated over $k$)?

(This is true for $X$ smooth by work of Florian Ivorra -- "Realisation l-Adique des Motifs Triangules Geometriques, I." and using deJong's resolutions and Galois descent, I suspect this should be true for all $X$. However, there is a issue related to "functoriality" of the cone which seems to be an obstacle for deducing this directly in the triangulated categories and I am stuck).