4 problems

Sharpness problem. For every $p>0$ and $q\geq 0$,

Sharpness for fusion systems. For all $i\geq 1$ and $j\geq 0$, the higher limits vanish:

Strong form of Oliver's conjecture.

The module-theoretic reformulation. If $V$ is an $F$-module, then there exists $1\neq g\in\Omega_1(Z(G))$ such that the minimal polynomial of the action of $g$ on $V$ divides