Generalized Purely Wild Inertia Conjecture

Let $G$ be a finite quasi $p$-group, let $r\geq 1$, and let $P_1,\ldots,P_r$ be non-trivial $p$-subgroups of $G$. Write $P_i^G$ for the set of conjugates of $P_i$ in $G$, and assume

$$G=\langle P_1^G,\ldots,P_r^G\rangle.$$

Let $B=\{x_1,\ldots,x_r\}$ be a set of closed points in $\mathbb{P}^1$.

Generalized Purely Wild Inertia Conjecture. There is a connected $G$-Galois cover of $\mathbb{P}^1$ étale away from $B$ such that $P_i$ occurs as an inertia group above $x_i$ for $1\leq i\leq r$.

This is the purely wild several-point specialization of the Generalized Inertia Conjecture and generalizes the Purely Wild Inertia Conjecture. The paper reports affirmative results for products of groups without a common quotient when the conjecture holds for each factor, but the general conjecture remains open.