Let $k$ be an algebraically closed field. Let $S$ be a homotopy invariant $\mathbb{Q}$-linear sheaf with transfers in the sense of Voevodsky–Suslin, and assume that the dimension of $S(U)$ (over $\mathbb{Q}$) is bounded for a constant $c$ if $U$ is a smooth (connected) $k$-variety. Is it known that $S$ is constant?


I probably have a somewhat clumsy proof of this fact using Suslin rigidity-type arguments; yet I wonder which facts related to this one are already known. Moreover, I am actually interested in the extension of $S$ to pro-smooth (say, affine) $k$-schemes; and my finite dimensionality assumption corresponds to the finite dimensionality of $S(\operatorname{Spec} K)$, where $K$ is an algebraically closed field extension of $k$ of infinite transcendence degree. Consequently, I would also like to know which statements should I cite to deal with "colimit extensions" of this sort; is this section 8.13 of EGA 4?



Are "strongly finite dimensional" homotopy invariant sheaves with transfers (locally) constant?
I have proved this statement as Lemma 5.1.3 at [Bondarko and Sosnilo - On Chow-weight homology of geometric motives](https://www.researchgate.net/publication/340849991_On_Chow-weight_homology_of_geometric_motives). Comments are very welcome!


