Radical-polynomial sectional volume conjecture for convex bodies

Let $K \subset \mathbb R^n$ be a compact body with $C^{\infty}$ boundary $\partial K$. For $\xi \in \mathbb R^n$ with $|\xi|=1$ and $t \in \mathbb R$, let

$$A_K(\xi,t)=\operatorname{vol}_{n-1}\bigl(K\cap\{x\in\mathbb R^n:\langle \xi,x\rangle=t\}\bigr)$$

be the sectional volume function. Radical-polynomial sectional volume conjecture. If, for some $m\in\mathbb N$, the $m$-th power $A_K^m(\xi,t)$ is a polynomial in $t$ whenever $A_K(\xi,t)\ne 0$, then $\partial K$ is an ellipsoid. This conjecture seeks to characterize ellipsoids by the algebraic dependence of their sectional volume functions; polynomially integrable domains with smooth boundary are known to be ellipsoids in odd dimensions, while the radical-polynomial formulation is intended to cover all dimensions.