The $p$-adic Tate conjecture for crystalline cohomology

Let $X$ be a smooth projective variety over the finite field $k=\operatorname{\mathbb{F}}_q$ of characteristic $p$, with $q=p^m$. Let $W(k)$ be the ring of $p$-typical Witt vectors, let $K=W(k)[1/p]$, and let $H^{\ast}_{\operatorname{crys}}(X):=H^{\ast}_{\operatorname{crys}}(X/W(k))\otimes_{W(k)}K$. Let $\operatorname{Fr}_p$ denote crystalline Frobenius. $p$-adic Tate conjecture. The cycle class map

$$\operatorname{CH}^{\ast}(X)\otimes\operatorname{\mathbb{Q}}_p\to {H^{2\ast}_{\operatorname{crys}}(X)(\ast)}^{\operatorname{Fr}_p}$$

with values in the $\operatorname{\mathbb{Q}}_p$-vector subspace fixed by $\operatorname{Fr}_p$ is surjective. This is the crystalline formulation of the Tate conjecture over finite fields and remains open in general.