Friedlander's conjecture on the K-theoretic Chern character

Let $\mathbf{F}=\mathbf{F}_q$ be a finite field, let $X$ be a smooth variety over $\mathbf{F}$, let $\bar X=X\times_{\mathbf{F}}\overline{\mathbf{F}}$, and let $G=\operatorname{Gal}(\overline{\mathbf{F}}/\mathbf{F})$. For integers $n$ and $i$, let $K_{2n-i}(X)_{\mathbf{Q}}^{(n)}$ denote the $n$-th Adams eigenspace in the rational algebraic $K$-theory of $X$. Friedlander's conjecture. The Chern character

$$ch_{n,i}:K_{2n-i}(X)_{\mathbf{Q}}^{(n)}\otimes\mathbf{Q}_l\longrightarrow H^i(\bar X,\mathbf{Q}_l(n))^G$$

is an isomorphism. This extends the preceding cycle-class statement from smooth projective varieties to smooth varieties and is attributed in the text to Friedlander; the supplied text gives no resolution status.