Let $k$ be a subring of $\mathbb{C}$. By the *$k$-Hodge conjecture*, we mean the statement that, for each nonsingular algebraic variety $X$ over $\mathbb{C}$, and each $p = 0, 1, \ldots, \dim_{\mathbb{C}}(X)$,  each class $\mathfrak{z} \in H^{2p}(X; k) \cap H^{p,p}(X)$ is a $k$-linear combination of classes of algebraic cycles. The millenium problem is the $\mathbb{Q}$-Hodge conjecture. Atiyah-Hirzebruch [proved in 1961][1] that the $\mathbb{Z}$-Hodge conjecture is false. Given $\mathbb{Z} \subset k \subset K \subset \mathbb{C}$, the $k$-Hodge conjecture implies the $K$-Hodge conjecture.

>For which $\mathbb{Z} \subset k \subset \mathbb{C}$ is the status of the $k$-Hodge conjecture known? 

More specifically, is the $k$-Hodge conjecture known to be true for some $\mathbb{Q} \subset k \subset \mathbb{C}$? In particular, is the $\overline{\mathbb{Q}}$-Hodge conjecture true?

In another direction, is the $k$-Hodge conjecture known to be false for some $\mathbb{Z} \subset k \subset \mathbb{Q}$?

This will be community wiki, so there can be one answer for each $k$.


  [1]: http://hirzebruch.mpim-bonn.mpg.de/151/1/29_Analytic%20cycles%20on%20complex%20manifolds.pdf