Purely Wild Inertia Conjecture

Let $G$ be a finite quasi $p$-group, meaning that $G$ is generated by its Sylow $p$-subgroups. Let $P$ be a $p$-subgroup of $G$.

Purely Wild Inertia Conjecture. The subgroup $P$ occurs as the inertia group at a point above $\infty$ in a connected $G$-Galois cover of $\mathbb{P}^1$ branched only at $\infty$ if and only if the conjugates of $P$ generate $G$.

This is the purely wild special case of the Inertia Conjecture. The paper refers to it as an established conjecture in the groups for which the purely wild part of the Inertia Conjecture is known; no general resolution is supplied here.