Grothendieck's variational Hodge conjecture

Let $k$ be a field of characteristic $0$, let $f:{\mathcal X}\to{\mathcal S}$ be a smooth projective morphism with ${\mathcal S}/k$ a smooth variety, and fix a point $s\in{\mathcal S}$ with fiber ${\mathcal X}_s$. For $\xi_s\in K_0({\mathcal X}_s)_{\mathbb Q}$, consider its de Rham Chern character ${\rm ch}(\xi_s)\in H^*_{\rm dR}({\mathcal X}_s/s)$.

Variational Hodge conjecture. The following are equivalent:

1. ${\rm ch}(\xi_s)$ lifts to an element of $H^*_{\rm dR}({\mathcal X}/k)$.
2. There is an element $\xi\in K_0({\mathcal X})_{\mathbb Q}$ such that

$${\rm ch}(\xi|_{{\mathcal X}_s})={\rm ch}(\xi_s).$$

This is Grothendieck's original, global formulation of the principle of parallel transport of algebraic cycles. The paper explains that it is equivalent to the infinitesimal formulation above; its resolution is not supplied here and the conjecture remains open.