Let $F$ be a number fields. Conjecturally, there is a rigid $\mathbb{Q}$-linear abelian category of mixed motives over $F$. Let $\mathbb{1}$ denotes the unit object of this category. Given a mixed motive $M$, we can consider the extension group $\operatorname{Ext}^1_{F}(\mathbb{1},M)$ which is a vector space over $\mathbb{Q}$. According to Scholl, we can define a subspace $\operatorname{Ext}^1_{\mathcal{O}_F}(\mathbb{1},M)$ of extensions having everywhere good reduction. Scholl conjectures (loc. cit.) that $\operatorname{Ext}^1_{\mathcal{O}_F}(\mathbb{1},M)$ is finite dimensional.

- The conjecture is stated inside a paragraph in Scholl's paper, so it seems that it comes from elsewhere. Was Scholl the first to formulate this conjecture?
- What are the arithmetic consequences of such a conjecture?
- For instance, does the finite generation of class groups, the fact that $\mathcal{O}_F^{\times}$ is finitely generated, or the Mordel-Weil Theorem follow from it? Are there more consequences?
- More generally, is the conjecture equivalent to the finite generation of some algebraic $K$-groups?

- I believe that, in most cases, $\operatorname{Ext}^1_{F}(\mathbb{1},M)$ is not finitely generated. Is this known or at least conjectured somewhere?

Sorry for keeping this a bit vague. Many thanks in advance!