The one-dimensional transverse Killing quotient conjecture for locally homogeneous pp-waves

Let $({\cal M},\mathrm{g})$ be an indecomposable locally homogeneous pp-wave of dimension greater than $3$. Let $V$ denote the parallel null vector field of the pp-wave, let $\mathfrak{k}$ be its Lie algebra of Killing vector fields, and let $\mathfrak{k}(V)$ be the subalgebra consisting of Killing vector fields tangent to $V^\perp$. The quotient $\mathfrak{k}/\mathfrak{k}(V)$ measures Killing directions not tangent to $V^\perp$. The one-dimensional transverse Killing quotient conjecture. The Lie algebra quotient $\mathfrak{k}/\mathfrak{k}(V)$ is one-dimensional. For strongly indecomposable plane waves, the analogous upper bound is proved in the source; the conjecture asks whether it remains true for arbitrary indecomposable locally homogeneous pp-waves in dimensions greater than $3$.