Projective involution symmetry conjecture for 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 $K$. Projective involution conjecture. If for every $X\in\mathcal S$ there is a different $Y\in\mathcal S$ such that the sections of $K$ by $X$ and $Y$ differ by a non-trivial projective involution, then $K$ and $\mathcal S$ are both invariant under the same non-trivial projective involution. This is posed as a projective generalization of the corresponding central-projection result and remains open.