Gusakova–Zaporozhets conjecture on ellipsoids and intrinsic volumes

Let $\mathcal E_1$ and $\mathcal E_2$ be two ellipsoids in $\mathbb R^n$. For an ellipsoid $\mathcal E$, let $V_i(\mathcal E)$ denote its $i$-th intrinsic volume. Gusakova–Zaporozhets conjecture. If

$$V_1(\mathcal E_1)=V_1(\mathcal E_2),\quad V_2(\mathcal E_1)=V_2(\mathcal E_2),\quad\ldots,\quad V_n(\mathcal E_1)=V_n(\mathcal E_2),$$

then $\mathcal E_1$ and $\mathcal E_2$ are congruent. The conjecture asks whether an ellipsoid is uniquely determined, up to an isometry, by its intrinsic volumes or, equivalently, by its Steiner polynomial. It was confirmed in $\mathbb R^3$ by Petrov and Tarasov; the supplied source does not state a resolution in general dimension.