Local abundance of flat shadow boundaries implies an ellipsoid

Let $K\subset\mathbb R^n$, $n\geq 3$, be a convex body with boundary of class $C^3$. Local light-source conjecture. If for every $p\in\partial K$ there are at least $n+1$ point light sources on the tangent hyperplane $T_p\partial K$, in general linear position with respect to $p$, that create flat shadow boundaries on $K$, then $K$ is an ellipsoid. The conjecture is motivated by a result requiring sufficiently many generic light sources and by the counterexample showing that $n$ sources do not suffice; it remains open.