Conjecture that planar grazes characterize ellipsoids

Let $K,L\subset\mathbb R^n$ be convex bodies with $L\subset\operatorname{int}K$. For a point $x\in\mathbb R^n\setminus L$, let $\Sigma(L,x)$ denote the graze of $L$ from $x$, namely the intersection of the boundary of the support cone of $L$ from $x$ with $\partial L$. Suppose that, for every $x\in\partial K$, $\Sigma(L,x)$ lies in a hyperplane. Planar-graze characterization conjecture. Then $L$ is an ellipsoid. The source presents this as an expected extension of known characterizations of ellipsoids by planar shadow boundaries and grazes; its status is not resolved in the supplied text.