The $p$-adic variational Hodge conjecture

Let $\mathcal{X}$ be a smooth proper $W(k)$-scheme, where $k$ is a perfect field of characteristic $p>0$. Suppose that $Z_0$ is a codimension $n$ cycle in $\mathcal{X}\times_{W(k)} k$ such that its cycle class

$$cl(Z_0)\in H_{\rm cris}^{2n}(\mathcal{X}\times_{W(k)}k/W(k))\otimes \operatorname{Frac}(W(k))\simeq H_{\rm dR}^{2n}(\mathcal{X}/W(k))\otimes\operatorname{Frac}(W(k))$$

lies in the $n$-th piece of the Hodge filtration. The $p$-adic variational Hodge conjecture. Then, there exists a cycle $Z\subset \mathcal{X}$ which specializes to $Z_0$ in the rational Chow group. This predicts that a special-fibre cycle whose crystalline cycle class satisfies the appropriate Hodge-filtration condition lifts, at least rationally in the Chow group, to the smooth proper mixed-characteristic family. The statement is presented as a conjecture in the source; no resolution is supplied there.