Larman-point conjecture for convex bodies

Let $K\subset \mathbb{R}^{n}$, $n\geq 3$, be a convex body. A point $p\in\mathbb{R}^{n}$ is a Larman point of $K$ if, for every hyperplane $\Pi$ passing through $p$, the section $\Pi\cap K$ has an $(n-2)$-plane of symmetry. A Larman point is a revolution point if every such section has an $(n-2)$-plane of symmetry containing $p$. Larman-point conjecture. If $p$ is a Larman point of $K$ that is not a revolution point of $K$, then either $K$ is an ellipsoid or $K$ is a body of revolution. The conjecture seeks to characterize convex bodies from the symmetry of hyperplane sections through one point; the source focuses on this question with additional hypotheses and does not report a resolution.