Collinear supporting-cone flatness 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 $S$, and with $S$ star-convex with respect to the origin. Supporting-cone flatness conjecture. If for every $x\in S$ there is a different $y\in S$ such that the line $xy$ passes through the origin and the intersection of the boundaries of the supporting cones of $K$ with apexes at $x$ and $y$ is flat, then $K$ is an ellipsoid. This is posed as a further characterization problem for ellipsoids and remains open.