Conjecture on intersections of support cones characterizing homothetic ellipsoids

Let $L,K\subset \mathbb{R}^{n}$ be convex bodies, with $n\geq 3$, and suppose that $L\subset \operatorname{int}K$. For $x\in\mathbb{R}^{n}\setminus K$, let $S(L,x)$ denote the boundary of the cone generated by $L$ with apex $x$. Support-cone intersection conjecture. For every $x\in\operatorname{bd}K$, there exist $y\in\operatorname{bd}K$ and a hyperplane $\Pi$ such that

$$S(L,x)\cap S(L,y)=\Pi\cap\operatorname{bd}K.$$

Then $L$ and $K$ are homothetic ellipsoids. The source presents this as a problem motivated by an observation about intersections of support cones; the supplied parser gives no evidence that it has been resolved.