Strong Oort conjecture for cyclic-by-p groups

Let $k$ be an algebraically closed field of characteristic $p$. A finite group is cyclic-by-$p$ if it is an extension of a cyclic group of order prime to $p$ by a normal $p$-Sylow subgroup. Let $C_m$ denote a cyclic group of order $m$, let $D_{2p^n}$ denote the dihedral group of order $2p^n$, and let an Oort group for $k$ be a finite group $G$ such that every $G$-Galois cover of smooth projective $k$-curves lifts to characteristic zero.

Strong Oort conjecture. A cyclic-by-$p$ group $G$ is an Oort group for $k$ if and only if $G$ is of the form $C_m$, $D_{2p^n}$, or $A_4$, with the last case occurring only if $p=2$.

This conjecture proposes the converse to the paper's classification of cyclic-by-$p$ Oort groups. The supplied text does not state whether it has been resolved, so its status remains open in this record.