The support-cone graze characterization of ellipsoids

Let $K,L\subset\mathbb R^n$ be convex bodies with $L\subset\operatorname{int}K$. For each $x\in\partial K$, let $\Sigma(L,x)$ denote the graze of $L$ from $x$, namely the intersection of the support cone of $L$ with apex $x$ and $\partial L$. Support-cone graze conjecture. If, for every $x\in\partial K$, the set $\Sigma(L,x)$ lies in a hyperplane, then $L$ is an ellipsoid. This extends the known characterization using support cones with apices on a plane or shadow boundaries, and the conjecture is presented as unknown even when the outer surface is the boundary of a convex body containing $L$ in its interior.