Beilinson–Bloch injectivity conjecture for the higher Abel–Jacobi map

Let $k$ be a field of characteristic $0$ embedded in ${\mathbb C}$, and let $X$ be a smooth projective variety over $k$. Define the homologically trivial cycles and the $m$-th higher Abel–Jacobi map by

$$\operatorname{CH}^m_{\mathrm{hom}}(X)_{\mathbb Q}=\operatorname{CH}^m_{\mathrm{hom}}(X)\otimes_{\mathbb Z}{\mathbb Q}\longrightarrow J^{2m-1}(X)_{\mathbb Q},$$

where

$$J^{2m-1}(X)=H^{2m-1}(X,{\mathbb C})\big/\bigl(F^mH^{2m-1}(X,{\mathbb C})\oplus H^{2m-1}(X,{\mathbb Z}(m))\bigr).$$

Beilinson–Bloch conjecture. The $m$-th higher Abel–Jacobi map $AJ^m$ is injective. This is presented as another part of the Beilinson–Bloch conjecture for smooth projective varieties over number fields; its injectivity is open in general.