The KGB obstruction conjecture for local extensions with cyclic p-Sylow subgroup

Let $k$ be an algebraically closed field of characteristic $p$, and let $G$ be a finite group. A local $G$-extension is a finite $G$-Galois extension of formal power series rings over $k$. The KGB obstruction is the obstruction to lifting such an extension to characteristic zero arising from the KGB invariant. A group is a local Oort group if every local extension with that group lifts to characteristic zero.

KGB obstruction conjecture. The KGB obstruction is the only obstruction to the local lifting problem for local $G$-extensions where $G$ has cyclic $p$-Sylow subgroup. In particular, $D_{p^n}$ is a local Oort group when $p$ is odd.

This conjecture generalizes the known result for metacyclic groups and would settle the local lifting problem for the groups $D_{p^n}$ in odd characteristic. It is known when the $p$-Sylow subgroup of $G$ is ${\mathbb Z}/p$, but remains open in general.