Ehlers–Kundt conjecture for complete Ricci-flat pp-waves

Let $(\mathbb{R}^4,g)$ be a Ricci-flat pp-wave with metric

$$g=2\,du\,dv+H(u,\mathbf{x})\,du^2+dx^2+dy^2,$$

where $H\colon\mathbb{R}\times\mathbb{R}^2\to\mathbb{R}$. Ehlers–Kundt conjecture. The metric is geodesically complete if and only if, for every $u\in\mathbb{R}$, the function $\mathbf{x}\mapsto H(u,\mathbf{x})$ is a polynomial of degree at most two. This conjecture identifies gravitational plane waves as the most elementary globally regular source-free gravitational fields. The paper studies its failure in the impulsive case; the supplied context does not establish a general resolution status beyond presenting the claim and its impulsive failure.