Translative equal-volume property and Euclidean constant width

Let $\mathcal{K}_d$ be the family of convex bodies in $\mathbb{R}^d$. A convex body has the translative equal volume property when the relevant volume associated with every intersecting translate is constant, as defined earlier in the paper. A body has constant width in a Euclidean space when every pair of parallel supporting hyperplanes is separated by the same Euclidean distance.

Translative constant-width conjecture. If $d\geq3$ and some $K\in\mathcal{K}_d$ satisfies the translative equal volume property, then $K$ is a convex body of constant width in a Euclidean space.

In the plane, the analogous property is characterized by constant-width bodies in Radon norms. The cited fact that a normed space of dimension at least three whose planar sections are all Radon is Euclidean motivates this higher-dimensional conjecture, whose resolution is not given in the supplied text.