A generalization of the Oort conjecture

The Oort conjecture (now a theorem of Obus–Wewers and Pop) states that if $k$ is an algebraically closed field of characteristic $p$, then any cyclic branched cover of smooth projective $k$-curves lifts to characteristic zero. This is equivalent to the local Oort conjecture, which states that all cyclic extensions of $k[[t]]$ lift to characteristic zero. We generalize the local Oort conjecture to the case of Galois extensions with cyclic $p$-Sylow subgroups, reduce the conjecture to a pure characteristic $p$ statement, and prove it in several cases. In particular, we show that $D_9$ is a so-called local Oort group.