Finite-dimensionality conjecture for integral motivic extension spaces

Let $F$ be a number field, let $X$ be a smooth projective variety over $F$, and let $M=h^{i-1}(X)(n)$. Write $の\operatorname{Ext}^1_{\mathcal{O}_F}(\mathbbm{1},M)$ for the subspace of extensions whose localization at every finite place $\mathfrak{p}$ belongs to $\operatorname{Ext}^1_{\mathcal{O}_{\mathfrak{p}}}(\mathbbm{1}_{\mathfrak{p}},M_{\mathfrak{p}})$. Finite-dimensionality conjecture. The space $\operatorname{Ext}^1_{\mathcal{O}_F}(\mathbbm{1},M)$ has finite dimension over $\mathbb{Q}$. This expectation is presented as the starting point for Beilinson's conjectures concerning integral motivic cohomology. The supplied context does not indicate whether this conjecture has been resolved.