Spectral-tree and mother-body conjecture for high-order Heun operators

Let $\mathfrak d$ be a high-order Heun operator with simple roots of its leading coefficient $Q_k$, and let $\mu_n$ denote the spectral root-counting measures. Let $K_Q$ be the relevant spectral convex-hull region, and let $U_Q^{\rm ext}$ be the exterior potential determined by the operator's leading data. A positive mother body is a positive measure with at most one-dimensional support and the same exterior logarithmic potential as the prescribed measure.

Spectral-tree and mother-body conjecture. The measures $\mu_n$ converge weakly to a probability measure $\mu_Q$ depending only on $Q_k$. Its support is a finite planar tree contained in $K_Q$, and the leaves of the tree are exactly the roots of $Q_k$. Moreover, $\mu_Q$ is a positive mother body for the exterior potential $U_Q^{\rm ext}$.

The conjecture strengthens the established convergence results in the collinear case and explains why the averaged finite-band measure may have two-dimensional support while the spectral roots appear to accumulate on a thin tree. The general two-dimensional case remains open.