The conjecture on powers of sectional-volume functions

Let $K$ be a body in $\mathbb R^n$ with $C^{\infty}$ boundary, and let $A_K(\xi,t)$ denote its sectional-volume function. Suppose that for some $m\in\mathbb N$, the power $A_K^m(\xi,t)$ is polynomial in $t$.

Power-polynomial conjecture. Then $\partial K$ is an ellipsoid; consequently, one can take $m=1$ when $n$ is odd and $m=2$ when $n$ is even.

This would classify bodies whose sectional-volume functions become polynomial after taking a positive integral power. The statement is given without a resolution in the supplied text.