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)