Compact Lorentz examples with only timelike closed geodesics

Construct a compact Lorentz manifold whose closed geodesics are all timelike.

Background

The paper constructs complete compact Lorentz examples whose closed geodesics are exclusively spacelike or exclusively null, and proves that certain Heisenberg quotients admit both timelike and spacelike closed geodesics. It explicitly states that no example is known in which every closed geodesic is timelike, leaving the existence of such an example unresolved.