Refinement of Bezdek's conjecture for Larman points

Let $K\subset \mathbb{R}^{n}$ be a convex body with $n\geq 3$, and let $p\in\operatorname*{\int}K$. For every hyperplane $\Pi$ passing through $p$, suppose the section $\Pi\cap K$ has an $(n-2)$-plane of symmetry; then $p$ is a Larman point of $K$. If, moreover, every such section has an $(n-2)$-plane of symmetry containing $p$, then $p$ is a revolution point of $K$. Refinement of Bezdek's conjecture. Suppose that $p$ is a Larman point of $K$ which is not a revolution point of $K$. Then either $K$ is an ellipsoid or $K$ is a body of revolution. This refinement concerns the global shape forced by the distinction between Larman and revolution points; the supplied source gives no resolution status.