Mother-body conjecture for the spectral limit of high-order Heun operators

Let $\mathfrak d$ be a high-order Heun operator with Fuchs index one and generic leading coefficient $Q_k$. Let $Sp_n$ be the spectral polynomial, let $\nu_{\mathfrak d}$ be the averaged finite-band measure, and let $K_Q$ be the relevant exterior region. A mother body is a positive measure with at most one-dimensional support and the same exterior logarithmic potential, or equivalently the same exterior Cauchy transform, as a given measure.

Mother-body conjecture. The root-counting measures of $Sp_n$ converge weakly to a positive measure $\mu_{Q_k}$ depending only on $Q_k$. Its support is a finite planar tree $\Gamma_{Q_k}\subset K_Q$, whose leaves are the roots of $Q_k$. Moreover,

$$U^{\mu_{Q_k}}(t)=U^{\nu_{\mathfrak d}}(t)$$

in the exterior of $K_Q$; equivalently, $\mu_{Q_k}$ has the same exterior Cauchy transform as the averaged finite-band measure $\nu_{\mathfrak d}$. Thus $\mu_{Q_k}$ is a positive mother body for the exterior field determined by the finite-band averaging procedure.

This refines the main spectral-limit conjecture by distinguishing the averaged finite-band measure from the one-dimensional limiting zero measure. The claim is motivated by numerical tree-like supports and remains open in the genuinely complex, two-dimensional setting.