Gusakova–Zaporozhets uniqueness conjecture for ellipsoids

Let $\mathcal{E}_1$ and $\mathcal{E}_2$ be ellipsoids in $\mathbb{R}^n$. For a bounded convex set $K\subset\mathbb{R}^n$, let $V_0(K),\ldots,V_n(K)$ denote its intrinsic volumes, defined by the Steiner formula

$$\operatorname{Vol}(K+tB_n)=\sum_{k=0}^n\kappa_{n-k}V_k(K)t^{n-k},$$

where $B_n$ is the Euclidean unit ball, $\kappa_k=\pi^{k/2}/\Gamma(\frac{k}{2}+1)$, and $\operatorname{Vol}$ is $n$-dimensional volume. Two ellipsoids are congruent when one is obtained from the other by a rigid motion. Gusakova–Zaporozhets' conjecture. If

$$V_i(\mathcal{E}_1)=V_i(\mathcal{E}_2)$$

for every $i=1,2,\ldots,n$, then $\mathcal{E}_1$ and $\mathcal{E}_2$ are congruent. The conjecture asserts uniqueness of an ellipsoid up to rigid motions from its intrinsic volumes. It was conjectured by Anna Gusakova and Dmitry Zaporozhets in 2017 and is proved in the source paper for $n=3$, while the general-dimensional case is not resolved here.