Conjecture on root-counting measures for generalized Heun equations

Consider a generalized Heun equation

$$\left\{Q_{k+1}(z)\frac{d^k}{dz^k}+Q_k(z)\frac{d^k}{dz^{k-1}}+\cdots+Q_2(z)\frac{d}{dz}+V(z)\right\}S(z)=0,$$

where $\deg Q_{k+1}(z)=k+1$ and $\deg Q_i(z)\leq i$ for $i=2,3,\ldots,k$. For each sufficiently large positive integer $n$, let $\mu_n$ be the root-counting measure of the corresponding spectral polynomial. Write $Conv_{Q_{k+1}}$ for the convex hull of the roots of $Q_{k+1}(z)$. Conjecture on generalized Heun root-counting measures. For any generalized Heun equation, the sequence $\{\mu_n\}$ converges to a probability measure $\mu$ supported on a curvilinear planar tree inside $Conv_{Q_{k+1}}$, whose leaves, namely its vertices of valency $1$, are the roots of $Q_{k+1}(z)$. Moreover, the limiting measure $\mu$ depends only on $Q_{k+1}(z)$ and is independent of the other coefficients of the equation. The claim is motivated by large-scale numerical experiments and extends the preceding root-distribution expectations from Heun equations to higher-order generalized Heun equations; no proof or resolution is given.