A new result [CS06] settles the practical aspects of the problem, but leaves one question unresolved. This paper provides an explicit algorithm that extends a quadrilateral surface mesh to a hexahedral mesh, where all the hexahedra have straight segment edges. In a sense, these hexahedra are intermediate between the topological and geometric meshes mentioned above. The faces are not necessarily planar, but this is not a crucial aspect in applications, such a fluid dynamics simulations.
The question of whether a hexahedral mesh with planar faces exists remains open.