Let $X$ be a smooth projective variety over an algebraically closed field $k$ of characteristic $p>0$. Let $W=W(k)$ be the ring of Witt vectors of $k$.

Assume that the crystalline cohomology $H^2_{crys}(X/W)$ is a torsionfree $W$-module.

Question. Does it follow that $H^2_{et}(X,\mathbb Z_\ell)$ is a torsionfree $\mathbb Z_\ell$-module for all prime numbers $\ell$ invertible in $k$?

Does a positive answer to this question imply that $NS(X)$ is torsionfree if it has no non-trivial $p$-torsion?

Let $X$ be a smooth projective variety over an algebraically closed field $k$ of characteristic $p>0$. Let $W=W(k)$ be the ring of Witt vectors of $k$.

Assume that the crystalline cohomology $H^2_{crys}(X/W)$ is a torsionfree $W$-module.

Question. Does it follow that $H^2_{et}(X,\mathbb Z_\ell)$ is a torsionfree $\mathbb Z_\ell$-module for all prime numbers $\ell$ invertible in $k$?

Do we "expect" the answer to this question to be positive or negative (assuming some standard conjectures)?