Let $k$ be an algebraically closed field of positive characteristic $p$ and fix a prime $\ell\neq p$. Let $X$ be a smooth connected $k$-variety and $f:Y\rightarrow X$ a smooth projective morphism. Assume that there is an open embedding with dense image $Y\rightarrow Y^{cmp}$ of $Y$ inside a smooth projective $k$-variety $Y^{cmp}$.

Is it true that the natual map $$H^{i}(Y^{cmp},\mathbb Q_{\ell}(j))\rightarrow H^{0}(X,R^{i}f_*\mathbb Q_{\ell}(j))$$ is surjective?

It seems like some cases are dealt by Deligne in Weil II (6.2.11), but I don't see if it's possible to extend it to this situation.