Ellipsoidal separation conjecture for totally positive matrix families

Let $\mathcal{B}_n, \mathcal{V}_n, \mathcal{C}_n \subset \mathbb{R}^{n \times n}$ denote the Bidiagonal-product, Vandermonde, and Cauchy totally positive matrix families generated by the random models of Section~. Let $\pi: \mathbb{R}^{n \times n} \to \mathbb{R}^3$ map a matrix $A$ to the three highest-order coefficients $(a_{n-1},a_{n-2},a_{n-3})$ of its characteristic polynomial. For a level $\alpha\in(0,1)$ and one of these families, let $E^{\alpha}(n)\subset\mathbb{R}^3$ be the Mahalanobis ellipsoid of the corresponding projected cloud enclosing a fraction $\alpha$ of its samples.

Ellipsoidal separation conjecture. For every fixed $\alpha\in(0,1)$, there exists a dimension $n_0(\alpha)$ such that, for all $n\ge n_0(\alpha)$, the three ellipsoids $E^{\alpha}_{\mathcal{B}}(n), E^{\alpha}_{\mathcal{V}}(n), E^{\alpha}_{\mathcal{C}}(n)$ are pairwise disjoint, and their pairwise separation, measured in units of the cloud spread, stays bounded below by a positive constant as $n$ grows. Equivalently, each family occupies its own convex, approximately ellipsoidal region of coefficient space, and these regions do not merge as the dimension increases.

The conjecture formalizes the observed increasing geometric separation among the three random totally positive matrix families in the space of their highest-order characteristic-polynomial coefficients. Establishing it would require asymptotic control of the projected centroids and covariances together with concentration bounds for $(a_{n-1},a_{n-2},a_{n-3})$; the supplied context presents this as the main open problem suggested by the experiments, with no resolution stated.