The Hodge 1-motive conjecture for proper complex schemes

Let $X$ be a proper scheme over $\mathbb C$, let $\pi:X_\bullet\to X$ be a smooth proper hypercovering, and let $H^{2p+i}(X)$ be Deligne's mixed $\mathbb Q$-Hodge structure on $H^{2p+i}(X,\mathbb Q)$. Let $NS^p$ and $A^p_{X_\bullet/\mathbb C}$ be the simplicial complexes of Néron–Severi groups and algebraic parts of intermediate Jacobians, let $\lambda_a^i$ be the associated boundary map, let $e^p$ be the extension-class map, and let $\Xi^{i,p}$ be the algebraically defined 1-motive. The Hodge 1-motive conjecture. The square formed by $\lambda_a^i$ and $e^p$ commutes, its motivic cycle-class map has image equal to the Hodge 1-motive of $H^{2p+i}(X)$, and

$$T_{\rm Hodge}(\Xi^{i,p})\cong H^{2p+i}(X)^h.$$

The conjecture proposes a canonical algebraic description of the Hodge 1-motive attached to mixed cohomology of a proper singular scheme; it is not established in the source.