The absolute Hodge conjecture for Hodge classes

Let $X$ be a smooth projective variety over an algebraically closed field $k=\overline{k}$ of finite transcendence degree over $\mathbb Q$. Let $(s_{\mathbb A_f},s_{\mathrm{dR}})$ be a tensor pair in

$$\bigl(H^w_{\mathrm{\acute et}}(X,\mathbb A_f)\times H^w_{\mathrm{dR}}(X/k)\bigr)^{\otimes}.$$

Absolute Hodge conjecture. If this pair comes from a Hodge class in $(H^w_B(X(\mathbb C),\mathbb Q))^{\otimes}$ for one embedding $\sigma:k\hookrightarrow\mathbb C$, then for every embedding $k\hookrightarrow\mathbb C$ it comes from a Hodge class under the comparison isomorphism with Betti cohomology. This conjecture asserts that Hodge classes are absolute Hodge classes. It is known in important cases, such as for abelian varieties by Deligne, but remains open for arbitrary smooth projective varieties.