Transitivity conjecture for Nil-Killing Lie algebras on CSI Lorentzian spaces

A Lorentzian space is CSI if all of its polynomial curvature invariants are constant. Let $M$ be an $n$-dimensional Lorentzian space, and let $\mathcal{N}$ denote a set of Nil-Killing vector fields on $M$. The set $\mathcal{N}$ is transitive when

$$\dim(\mathcal{N}|_p)=n$$

for every point $p\in M$.

Transitivity conjecture. If $M$ is an $n$-dimensional Lorentzian space with all curvature invariants constant, then there exists a transitive set $\mathcal{N}$.

This conjecture proposes that CSI Lorentzian spaces admit enough Nil-Killing vector fields to act transitively. The source gives this as an expectation based on known CSI examples, but provides no resolution in the stated passage.