The hierarchy conjecture for k-recurrent Lorentzian manifolds

Let $D\subset \mathcal V$ be a simply connected domain of an $n$-dimensional $k$-recurrent Lorentzian manifold $(\mathcal V,g)$. A vector field is null recurrent if it is null and recurrent. Hierarchy conjecture. If there is no null recurrent vector field on $D$, then $(D,g)$ is recurrent. The paper proposes this as a further extension of the hierarchy suggested for $k$-symmetric spacetimes; it is stated as an open conjecture, with no resolution given in the source.