Ellipsoid characterization from the translative constant volume property

Let $n\geq 3$, and let $K\in\mathcal{K}_n$ be an origin-symmetric convex body satisfying the translative constant volume property.

Translative constant volume conjecture. Then $K$ is an ellipsoid.

The source presents this as plausible for centrally symmetric bodies. It notes that the corresponding characterization is known in the planar setting in terms of constant width in a Radon plane, while the higher-dimensional centrally symmetric case remains open.