The Grothendieck period conjecture for de Rham-Betti classes

Let $X$ be a smooth projective variety defined over $\overline{\mathbb{Q}}$, and let $0\leq j\leq\dim(X)$ be an integer. Consider the de Rham-Betti structure

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

If $v\in V_{\mathrm{B}}$ satisfies $\rho'_{m}(v\otimes 1)\in V_{\mathrm{dR}}\otimes 1$, call $v$ a de Rham-Betti class on $X$. Grothendieck period conjecture. Every de Rham-Betti class on $X$ comes from a $\mathbb{Q}$-coefficient algebraic cycle on $X$. This is presented as a version of the Grothendieck period conjecture and is one of the motivations for the theory of de Rham-Betti structures; its general status is open.