2 problems

Nagel conjectures. For appropriate values of $C,D$, equality holds in the eventual linear bound for regularity, and there exist integers $E,F$ such that

Fixed-point control conjecture. If $I$ is a $\Gamma$-invariant faithful prime ideal of $kG$, then $I$ is controlled by the subgroup of $\Gamma$-fixed points $G^\Gamma$: