The indecomposable locally homogeneous pp-wave conjecture

Let $({\cal M},\mathrm{g})$ be an indecomposable locally homogeneous pp-wave of dimension greater than $3$. A locally homogeneous pp-wave is a pp-wave whose local Killing vector fields span the tangent space at every point, and indecomposable means that its metric does not locally split as an orthogonal product. A plane wave is a pp-wave whose profile function is quadratic in the transverse coordinates. The indecomposable locally homogeneous pp-wave conjecture. Every indecomposable locally homogeneous pp-wave of dimension greater than $3$ is a plane wave. This is motivated by examples showing that locally homogeneous pp-waves which are not plane waves can be decomposable, while the corresponding assertion in dimension $3$ requires a curvature-rank condition that cannot generally be omitted.