The finite generation conjecture for motivic cohomology of arithmetic schemes

Let $F$ be a number field and let $\mathcal{X}/\mathcal{O}_F$ be a smooth projective scheme of Krull dimension $d$. Write $H^i(\mathcal{X}_{\operatorname{\acute et}},\mathbb{Z}(n))$ for the étale motivic cohomology group defined using Bloch's cycle complex for $n\geq 0$ and the specified complexes for $n<0$.

Finite generation conjecture. For every $n\in\mathbb{Z}$ and every $i\leq 2n+1$, the group

$$H^i(\mathcal{X}_{\operatorname{\acute et}},\mathbb{Z}(n))$$

is finitely generated.

This expectation generalizes classical finiteness results in number theory, including finite generation of units and class groups and the Mordell–Weil theorem. The supplied text presents it as an optimistic expectation and gives no evidence of a resolution.