Two-body flat-shadow characterization of concentric homothetic ellipsoids

Let $K,L\subset\mathbb R^n$, $n\geq 3$, be convex bodies with $L\subseteq K$. Two-body flat-shadow conjecture. If for every $p\in\partial K$ there are at least $L(n)$ point light sources on the supporting cone of $L$ with apex at $p$ such that $L$ casts a shadow with flat boundary on $K$, then $K$ and $L$ are concentric, homothetic ellipsoids. This generalizes a theorem for light sources at boundary points and parallel shadow hyperplanes; the stated generalization remains open.