Generalized Inertia Conjecture

Let $r\geq 1$ be an integer, let $G$ be a finite quasi $p$-group, and for each $1\leq i\leq r$ let $I_i$ be a subgroup of $G$ that is an extension of a $p$-group $P_i$ by a cyclic group of order prime to $p$. 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 Inertia Conjecture. There is a connected $G$-Galois cover of $\mathbb{P}^1$ étale away from $B$ such that $I_i$ occurs as an inertia group above $x_i$ for $1\leq i\leq r$.

This generalizes the one-point Inertia Conjecture to prescribed inertia groups at several points. The paper notes partial results for certain cases, including levels $l\in\{0,1\}$, but the conjecture is not settled in general.