The centroid-section conjecture for pairs of convex bodies

Let $K_1,K_2\subset\mathbb R^n$, $n\geq 2$, be different convex bodies. For every affine hyperplane $H\in\operatorname{Graff}_{n-1}(\mathbb R^n)$ that intersects both bodies, consider the centroids of $K_1\cap H$ and $K_2\cap H$. Centroid-section conjecture. If these centroids coincide for every such $H$, then $K_1$ and $K_2$ are concentric, homothetic ellipsoids. The conjecture was recently proved by F. Nazarov, D. Ryabogin, V. Yaskin, and B. Zawalski for centrally symmetric convex bodies with boundary of class $C^1$; the unrestricted statement remains unresolved.