Nagata's theorem extension and the Grothendieck–Remak property for tensor categories

Let $k$ be a field and let $\mathcal{C}$ be a tensor category of moderate growth. Write $\mathsf{Alg}\mathcal{C}$ for its category of algebra objects, and let $A^0$ denote the maximal ordinary $k$-algebra associated with $A\in\mathsf{Alg}\mathcal{C}$. A tensor category is GR when it has the Grothendieck–Remak property described in Theorem (1), namely finite generation of $A^0$ for every finitely generated $A$ implies the corresponding finite-generation conclusion for $\mathcal{C}$. Nagata's extension conjecture. Nagata's theorem extends from tannakian categories to arbitrary tensor categories of moderate growth: the implication in Theorem (1) is an if and only if. The same conjecture asserts that the category $\mathsf{Ver}_{p^\infty}$ is GR. This proposes a generalization of Nagata's finite-generation theorem beyond tannakian categories; the source provides no resolution status for either assertion.