The Riemann hypothesis for varieties over a finite field has been proven by Deligne. Still I would like to ask the following question.
A variety $X$ over a finite field $k$ is liftable if there exists a number field $K$ and a variety $\mathcal X$ over $K$ such that $X$ is the reduction of $\mathcal X$ (with respect to some model) at some maximal ideal of $O_K$.
Suppose RH holds for liftable varieties (forgetting Deligne proved this). Can we deduce RH for all varieties from this?