The star-surface support-cone characterization of ellipsoids

Let $S\subset \mathbb{R}^{n}$ be an embedding of $\mathbb{S}^{n-1}$ in $\mathbb{R}^{n}$ that is $O$-star, and let $\phi:S\to S$ map each point to the unique point on the ray from $O$ in the opposite direction. Let $K\subset \mathbb{R}^{n}$ be a convex body with $n\geq 3$, $O\in \operatorname{int}K$, and $K\subset \operatorname{int}S$. The star-surface support-cone conjecture. If, for every $x\in S$, there exists a hyperplane $\Pi(x)$ containing $O$ such that

$$S(K,x)\cap S(K,\phi(x))\subset \Pi(x),$$

then $K$ is an ellipsoid. The preceding theorem establishes the analogous assertion for an $O$-symmetric ellipsoid, but the general convex-body problem is presented as an open problem.