The flat-graze conjecture for convex bodies

Let $M,K$ be convex bodies in $\mathbb{R}^n$ with $M\subset\operatorname{int}K$ and $n\geq 3$. For $x\in\operatorname{bd}K$, let the graze $\Sigma(M,x)$ be the set of contact points on support lines to $M$ through $x$. Flat-graze conjecture. If $\Sigma(M,x)$ is flat for every $x\in\operatorname{bd}K$, then $M$ is an ellipsoid.

This conjecture extends results of Marchaud and Burton, which establish the ellipsoid conclusion under related hypotheses involving a hyperplane or a neighborhood of the boundary. The stated formulation remains open.