Grothendieck–Hodge conjecture for algebraic 1-motives

Let $X$ be smooth and proper over $\mathbb C$. Let $\operatorname{gr}^0_{F_m}CH^p(X)$ denote the degree-zero graded piece of the motivic filtration on codimension-$p$ Chow groups, let $A^{p+1}_{X/k}$ be the algebraic part of the relevant intermediate Jacobian, and let $T_{\rm Hodge}$ denote Hodge realization. Grothendieck–Hodge conjecture.

$$T_{\rm Hodge}([\operatorname{gr}^0_{F_m}CH^p(X)\to 0]_{\mathbb Q})=H^{p,p}_{\mathbb Q}$$

and

$$T_{\rm Hodge}([0\to A^{p+1}_{X/k}]_{\mathbb Q})=(H^{p,p+1}+H^{p+1,p})_{\mathbb Q}.$$

This reformulates the conjecture as the assertion that algebraically defined 1-motives have precisely the expected Hodge realizations; the general assertion is open.