Ghys's conjecture on nilpotent subgroups of finite diffeomorphism groups

Let $M$ be a compact differentiable manifold, and let $G$ be a finite subgroup of the diffeomorphism group of $M$. Ghys's conjecture. There is a constant $I$ such that $G$ has a nilpotent subgroup of index at most $I$. This conjecture proposed a uniform bound on the algebraic complexity of finite group actions on each compact differentiable manifold, but it was disproved; Ghys subsequently modified it.