Thomason's Tate conjecture via $K$-theory localization

Let $X$ be a smooth projective variety over $\operatorname{\mathbb{F}}_q$, let $l$ be a prime, and for each positive integer $n$ consider $X\times_{\operatorname{\mathbb{F}}_q}\operatorname{Spec}\operatorname{\mathbb{F}}_{q^n}$. Say that an abelian group $A$ is $l$-reducible when $\operatorname{Hom}(\operatorname{\mathbb{Z}}(l^{\infty}),A)=0$. Let $L_{KU}K(-)$ denote Bousfield localization of the algebraic $K$-theory spectrum with respect to complex topological $K$-theory $KU$. Thomason conjecture. For all $n$, the group

$$\pi_{-1}L_{KU}K(X\times_{\operatorname{\mathbb{F}}_q}\operatorname{Spec}\operatorname{\mathbb{F}}_{q^n})$$

is $l$-reducible. Thomason showed that this statement is equivalent to the Tate conjecture for $X$ for all codimensions; its general status is therefore open.