Bianchi–Gruber conjecture on centrally symmetric sections

Let $K\subset\mathbb R^n$, $n\geq 3$, be a strictly convex body, and let $\mathcal S$ be a family of hyperplanes that is the image of an embedding of the tangent bundle of the sphere $\mathbb S^{n-1}$, with every hyperplane in $\mathcal S$ intersecting the interior of $K$. Bianchi–Gruber conjecture. If every such intersection is centrally symmetric and, with possibly a few exceptions, does not contain a possibly existing center of symmetry of $K$, then $K$ is an ellipsoid. The conjecture is known under the stronger assumption that all the sections are ellipsoids, while the stated version remains open.