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

Let $X$ be smooth and proper over a finite field, and let $H^i_{\mathcal M}(X,\mathbb Z(n))$, $H^i_F(X,\mathbb Z(n))$, $H^{i-1}_K(X,\mathbb Z(n))$, and $H^i_W(X,\mathbb Z(n))$ be the motivic, Weil–étale, and Kato cohomology groups occurring in the paper's exact diagram. Geisser's cohomological finite generation conjecture. The boldface groups in that diagram are finitely generated for all $i$, $n$, and $X$. Finite generation of motivic cohomology generalizes Bass's conjecture, while finite generation of Weil–étale cohomology is equivalent to Tate's and Beilinson's conjectures. The general assertion remains open.