The conjecture that algebraically integrable hypersurfaces are quadrics

Let $M$ be a smooth polynomially integrable hypersurface in Euclidean space, meaning that its local section-volume function depends polynomially on the displacement parameter. A strictly convex quadric is a strictly convex hypersurface defined by a quadratic equation.

Quadraticity conjecture. Every smooth polynomially integrable hypersurface is a strictly convex quadric in $\mathbb R^{2k+1}$.

Examples include ellipsoids, elliptic paraboloids, and single sheets of two-sheet hyperboloids. Polynomial integrability does not occur in even dimensions, but the asserted classification in odd dimensions is presented as an open problem.