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Barker–Larman Conjecture: Central symmetry of sections tangent to an inscribed sphere

Prove that for a convex body K ⊂ ℝ^n (n ≥ 3), if there exists a sphere B contained in the interior of K such that for every supporting hyperplane Π of B the section Π ∩ K is centrally symmetric, then K is an ellipsoid.

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Background

Another classical open problem cited by the authors is due to Barker and Larman. It concerns whether central symmetry of all sections of a convex body by hyperplanes supporting an inscribed sphere implies that the body is an ellipsoid.

The paper’s results on ellipsoidal sections tangent to a sphere are presented as steps toward addressing this conjecture.

References

... and, on the other hand, the J. A. Barker and D. G. Larman's Conjecture: Let $K \subset R$ be a convex body, $n \geq 3$. If there exists a sphere $B\subset int K$ such that, for every supporting hyperplane $\Pi$ of $B$, the section $\Pi \cap K$ is centrally symmetric, then $K$ is an ellipsoid.

— On characteristic properties of the ellipsoid in terms of circumscribed cones of a convex body (2401.03983 - Morales-Amaya et al., 8 Jan 2024) in Introduction