The hierarchy conjecture for k-symmetric Lorentzian manifolds

Let $D\subset \mathcal V$ be a simply connected domain of an $n$-dimensional $k$-symmetric Lorentzian manifold $(\mathcal V,g)$. A vector field is null covariantly constant if it is null and covariantly constant. Hierarchy conjecture. If there is no null covariantly constant vector field on $D$, then $(D,g)$ is locally symmetric. If true, this would yield a classification in which the $k$-symmetric spacetimes are direct sums of lower-order symmetric spaces and a particular $k$-symmetric spacetime of Brinkmann type; the conjecture is presented as an open problem for arbitrary $k$.