Abhyankar's inertia conjecture

Let $p$ be a prime, let $G$ be a finite quasi $p$-group, and let $I$ be a subgroup of $G$ that is an extension of a $p$-group by a cyclic group of order prime to $p$. Call $(G,I)$ realizable if there is a connected $G$-Galois cover of $\mathbb{P}^1$ branched only at infinity with inertia group $I$ at a point above infinity. For a $p$-subgroup $P$ occurring in $I$, write $P^G$ for its set of conjugates in $G$.

Abhyankar's inertia conjecture. The pair $(G,I)$ is realizable if and only if

$$G=\langle P^G\rangle.$$

The condition on $I$ is necessary from the theory of extensions of local fields. The conjecture is known in some cases, including when $G$ is a $p$-group, but its general status remains open.