The quadrics conjecture for locally polynomially integrable strictly convex hypersurfaces

Let $M$ be a strictly convex smooth connected hypersurface in $\mathbb R^n$, and let $\widehat M$ be its convex hull. For each $a\in M$, let $A_M^a(t)$ denote the $(n-1)$-dimensional volume of the section of $\widehat M$ by the hyperplane at signed distance $t$ from the tangent hyperplane at $a$. The hypersurface is locally polynomially integrable if, for every $a\in M$, there is an $\varepsilon_a>0$ such that $A_M^a(t)$ is polynomial for $t\in[0,\varepsilon_a)$. The quadrics conjecture. The only locally polynomially integrable strictly convex $C^{\infty}$ hypersurfaces are, up to an affine transformation, the listed quadrics in odd-dimensional spaces: ellipsoids, two-sheet hyperboloids, and elliptic paraboloids. Odd-dimensional quadrics provide examples, while the classification of all locally polynomially integrable strictly convex hypersurfaces is the main open problem addressed by the paper.