Liftability conjecture for non-split metacyclic actions on surfaces

Let $S_g$ be a closed orientable surface of genus $g$, and let a non-split metacyclic action mean an action of a finite metacyclic group on $S_g$ that is not split. A regular cyclic cover is a finite regular covering of surfaces whose deck transformation group is cyclic.

Liftability conjecture. Every non-split metacyclic action on $S_g$ lifts under a suitably chosen finite regular cyclic cover to a split metacyclic action.

The preceding proposition gives an explicit liftability criterion in terms of the associated data set, and the stated corollary shows that several families of dihedral-type metacyclic actions factor through split metacyclic actions. The conjecture asserts that an appropriate finite regular cyclic cover exists for every non-split metacyclic action.