Geisser's finite generation conjecture for motivic, Weil–étale and Kato cohomology

Let $X$ be a scheme over a finite field, and let $H_i^c(X,\mathbb Z(n))$, $H_i^F(X,\mathbb Z(n))$, and $H_i^K(X,\mathbb Z(n))$ denote the cycle, Weil–étale, and Kato homology groups introduced in the paper. Geisser's finite generation conjecture. The groups

$$H_i^c(X,\mathbb Z(n)),\qquad H_i^F(X,\mathbb Z(n)),\qquad H_i^K(X,\mathbb Z(n))$$

are finitely generated for every $i$, $n$, and $X$. This is the paper's main integral finite-generation conjecture; the finite-coefficient analogue is weaker, and the conjecture is related to finite generation questions for motivic and Weil–étale cohomology.