Once again on an analogue of the certain Voevodsky theorem

Suppose that $F$ is an ${\mathbb{A}}^1$-invariant quasi-stable $\mathbb{Z}{F}_{\ast }$-presheaf. Then its Zariski sheafification $F_{Zar}$ coincides with its Nisnevich sheafification $F_{Nis}$. Moreover, if $X\in Sm\mathrm{/}k$ is $k$-smooth, then for any $n$ there is equality $H_{Zar}^n(X,F_{Zar})=H_{Nis}^n(X,F_{Nis})$.