Central symmetry from inversely homothetic sections

Let $K,S\subset\mathbb R^n$, $n\geq 3$, be convex bodies with $S\subset\operatorname{int}K$. Inversely homothetic sections conjecture. If, for every pair of parallel hyperplanes $H_1,H_2$ supporting $S$, the sections $K\cap H_1$ and $K\cap H_2$ are inversely homothetic with respect to the point lying on the segment connecting the contact points of $K$ with $H_1$ and $H_2$, then $K$ and $S$ are centrally symmetric with respect to the same point. The statement is presented as a parallel-section counterpart of the projective involution conjecture and remains open.