Geodesic embedding for some geometrization of any finite surface action

Establish that for every finite group G of homeomorphisms of a closed surface S of genus greater than 1, there exists a hyperbolic metric on S making G act by isometries such that the pair (S,G) admits a totally geodesic embedding into a closed hyperbolic 3-manifold.

Background

The authors prove geodesic embeddability for all irreducible (triangle type) actions and for certain reducible actions (quadrangle type). They note that, in general, reducible actions have uncountably many geometric realizations while only countably many can embed, so not every fixed geometric realization will embed.

They conjecture that for any finite group action on a closed surface, at least one geometric realization (i.e., a choice of hyperbolic metric for which the group acts by isometries) should embed geodesically into a closed hyperbolic 3-manifold.