The dihedral weak local Oort conjecture in characteristic two

Let $D_{2^n}$ denote the dihedral group of order $2^n$, and let $k$ be an algebraically closed field of characteristic $2$. A group is a weak local Oort group if at least one local extension with that group lifts to characteristic zero.

Dihedral weak local Oort conjecture. The group $D_{2^n}$ is a weak local Oort group for $2$.

This conjecture is motivated by the proof that $D_4$ is weak local Oort. It extends that evidence to all dihedral $2$-groups, but the source gives no general proof.