Uniqueness of a 3D ellipsoid with given intrinsic volumes

Let $\mathcal{E}$ be an ellipsoid in ${\mathbb{R}}^n$. A. Gusakova and D. Zaporozhets conjectured that $\mathcal{E}$ is uniquely (up to rigid motions) determined by its intrinsic volumes. We prove this conjecture for $n=3$.