Given a presheaf $\mathcal{F}: Sm/k\rightarrow Ab$ we define a new presheaf $C\mathcal{F}= \varinjlim\limits_{X\times \{0,1\}\subset U \subset X\times \mathbb{A}^1}\mathcal{F}(U)$. The presheaf $\mathcal{F}$ is rationally contractible if there is a morphism $h:\mathcal{F}\rightarrow C(\mathcal{F})$ such that for the pullback functors $i_0^*, i_1^*$ along the sections $X\times \{0\}$ and $X\times \{1\}$, we have $i_1^*\circ h =id$ and $i_0^*\circ h =0$.

Given a rationally contractible presheaf is it possible to embed it into a rationally contractible injective presheaf? (This I think would imply the existence of rationally contractible injective resolution, since the quotient of two rationally contractible presheaves is rationally contractible)

Given a presheaf $\mathcal{F}: Sm/k\rightarrow Ab$ we define a new presheaf $C\mathcal{F}= \varinjlim\limits_{X\times \{0,1\}\subset U \subset X\times \mathbb{A}^1}\mathcal{F}(U)$. The presheaf $\mathcal{F}$ is rationally contractible if there is a morphism $h:\mathcal{F}\rightarrow C(\mathcal{F})$ such that for the pullback functors $i_0^*, i_1^*$ along the sections $X\times \{0\}$ and $X\times \{1\}$, we have $i_1^*\circ h =id$ and $i_0^*\circ h =0$.

Given a rationally contractible presheaf is it possible to embed it into a rationally contractible injective presheaf? If so is it possible to give a rationally contractible resolution?

Note: Even if this is possible it shouldn't be possible for the cohomology presheaves. For example motivic complexes are rationally contractible but their cohomology groups include Chow groups, which are not rationally contractible.

Motivation: An important class of complexes of sheaves satisfying this property are motivic complexes (different constructions have this property). This technical property turns out to be a crucial unique identifier of the motivic complexes. More precisely any family of complexes indexed by natural numbers like $C(n)$, which satisfy the properties of the Bloch-Ogus cohomology theory and the weight zero complex denoted by $C$ is the constant sheaf $\mathbb{Z}$, if furthermore we assume that the sheaves appearing in the complexes $C(n)$ are rationally contractible and the complex $C(n)$ is zero above degree $n$, this forces $C(n)$ to be the weight-$n$ motivic complex.