Geisser's finiteness conjecture for the groups $C_i(X)$

Let $X$ be smooth and projective over a finite field, and let $C_i(X)$ be the homology groups of the complex $KC_1^{F\operatorname{et}}(X)$ measuring the difference between Weil–étale and étale homology in weight $1$. Finiteness conjecture for $C_i(X)$. Every group $C_i(X)$ is finite. These groups measure the remaining discrepancy between the two cohomology theories in the weight-one case; the paper presents their finiteness as an open conjectural property.