Crystalline Variational Tate Conjecture

Let $f:X\to S$ be a smooth, proper morphism of smooth $k$-varieties, where $k$ is a perfect field of characteristic $p>0$, let $s\in S$ be a closed point, and let $z\in \operatorname{CH}^i(X_s)_\mathbb{Q}$. Write

$$c:=cl_i(z)\in H^{2i}_{\operatorname{crys}}(X_s).$$

Here $H^n_{\operatorname{crys}}(-)$ denotes rational crystalline cohomology, $cl_i$ is the crystalline cycle class map, $\phi$ is absolute Frobenius, and $H^0_{\operatorname{crys}}(S,R^{2i}f_*\mathcal{O}_{X/K})$ is the space of flat crystalline sections. Crystalline Variational Tate Conjecture. The following conditions are equivalent: there exists $\widetilde z\in \operatorname{CH}^i(X)_\mathbb{Q}$ such that $cl_i(\widetilde z)|_{X_s}=c$; $c$ lifts to $H^{2i}_{\operatorname{crys}}(X)$; $c$ lifts to $H^{2i}_{\operatorname{crys}}(X)^{\phi=p^i}$; and $c$ is flat, meaning that it lifts to $H^0_{\operatorname{crys}}(S,R^{2i}f_*\mathcal{O}_{X/K})$. The conjecture is a crystalline characteristic-$p$ analogue of Grothendieck's Variational Hodge and $\ell$-adic Tate conjectures. It is proved in the source for divisors, together with an infinitesimal variant for higher-codimension cycles, but the full equivalence for arbitrary codimension is not established there.