Tannakian reformulation of Matzat's freeness conjecture

Let $k$ be an algebraically closed field of characteristic zero, set $K=k(x)$ with derivation $d/dx$, and let $F_X$ be the abstract free group on a set $X$ of cardinality $|k|$. A finite-dimensional $k$-linear representation of $F_X$ is cofinite if all but finitely many elements of $X$ act trivially. Let the category of finite-dimensional differential modules over $K$ and the category of all cofinite representations of $F_X$ be regarded as neutral tannakian categories over $k$. Tannakian formulation of Matzat's conjecture. The tannakian category of finite-dimensional differential modules over $K$ is equivalent to the tannakian category of all cofinite representations of $F_X$.

This is a tannakian reformulation of Matzat's freeness conjecture, using the identification of the absolute differential Galois group with the fundamental group scheme of the differential-module category. It is therefore equivalent in content to the freeness assertion above; the supplied text gives no separate resolution beyond the countable-transcendence-degree case proved in the paper.