The finite-filtration generation conjecture for centers and derived subgroups

Let $\operatorname{Aut}^0(X)=\lim A_n$ be an admissible filtration by irreducible affine subvarieties. Let $G\subseteq\operatorname{Aut}^0(X)$ be nilpotent or solvable and suppose that $G=\langle G\cap A_n\rangle$ for some $n$. Write $z(G)$ for the center of $G$. The finite-filtration generation conjecture. If $G$ is nilpotent, then $z(G)=\langle z(G)\cap A_m\rangle$ for some $m$; respectively, if $G$ is solvable, then $(G,G)=\langle (G,G)\cap A_m\rangle$ for some $m$. The source presents this as a sufficient condition for the induction toward the a-generated algebraicity conjecture, with no resolution stated.