Termination conjecture for the motivic filtration on higher Chow groups

Let $X$ be a variety, let $r$ and $m$ be integers, and let $F^\bullet{\mathrm{CH}}^r(X,m)$ denote the filtration on the higher Chow group. The source states that termination is equivalent to the injectivity assertion below:

$$\rho^{r}_X:F^r{\mathrm{CH}}^r(X,m)\longrightarrow {\mathrm{Ext}}^{r}_{\underline{{\mathrm{M}}}({\bold C})}({\bold Q}(0),H^{r-m}(X)(r)).$$

Termination conjecture for the motivic filtration. The filtration terminates, meaning

$$F^{N}{\mathrm{CH}}^r(X,m)=0$$

for some $N\gg 0$; equivalently, $F^{r+1}{\mathrm{CH}}^r(X,m)=0$, or the displayed higher Abel-Jacobi map is injective for each $X$, $r$ and $m$. This is presented as a conjecture about the filtration introduced in the paper. The source gives no resolution status.