Plane-wave conjecture for reductive Lorentzian homogeneous spaces

Let $G/H$ be a reductive Lorentzian homogeneous space of dimension $m\ge 4$, with isotropy group $H$ acting indecomposably but non-irreducibly. A plane wave is a Lorentzian manifold admitting a parallel null vector field, whose curvature tensor satisfies $R(X,Y)=0$ for all $X,Y$ orthogonal to that vector field, and such that $\nabla_XR=0$ for all $X$ orthogonal to it. Plane-wave conjecture. The homogeneous space $G/H$ is a plane wave. To our knowledge, all known examples satisfy this conclusion; the conjecture is motivated by the classification of homogeneous plane waves and by the analogous classification of Lorentzian symmetric spaces with indecomposable holonomy.