Grothendieck's de Rham-Betti cycle conjecture

Let $X$ be a smooth projective variety defined over $\overline{\mathbb{Q}}$. Consider the de Rham-Betti structure

(V_{\mathrm{B}},V_{\mathrm{dR}},\rho_{m}')=(\mathrm{H}^{2p}_{\mathrm{B}}(X,\mathbb{Q}),\mathrm{H}^{2p}_{\mathrm{dR}}(X/\overline{\mathbb{Q}),\rho_{m}) \otimes \mathbb{Q}_{\mathrm{dRB}}(p).

Grothendieck's de Rham-Betti cycle conjecture. Every de Rham-Betti class in $V_{\mathrm{B}}$ comes from a $\mathbb{Q}$-coefficient algebraic cycle on $X$ for $0\leq p\leq \dim(X)$. This conjecture proposes a de Rham-Betti analogue of the Hodge conjecture, relating classes defined by comparison data to algebraic cycles.