Flat shadow boundary characterization of ellipsoids

Let $K\subset\mathbb R^n$, $n\geq 3$, be a convex body and let $S$ be a hypersurface that is the image of an embedding of the sphere $\mathbb S^{n-1}$, with $K$ contained in the interior of $S$. Flat shadow boundary conjecture. If every point light source on $S$ creates a flat shadow boundary on $K$, then $K$ is an ellipsoid. This extends known results for polyhedral enclosing hypersurfaces and for point sources in an open neighborhood of $K$; the general case is presented as a central open problem.