WKB Stokes-period conjecture for the spectral tree of high-order Heun operators

Let $Q_k$ be a generic polynomial with simple roots. For a spectral parameter $t$, consider the algebraic differential

$$\omega_t=\left(\frac{z-t}{Q_k(z)}\right)^{1/k}dz,$$

with the branch normalized by $\omega_t=(z^{-1}+O(z^{-2}))dz$ at infinity, and let

$$\Pi_\gamma(t)=\int_\gamma\omega_t$$

be its period on the curve $y^k=(z-t)/Q_k(z)$. Let $\Gamma_{Q_k}$ be the support of the limiting spectral measure. For a cycle $\gamma$, let $E_\gamma$ be the set of parameters satisfying the real-period condition and having a positive corresponding WKB cut system.

WKB Stokes-period conjecture. The support $\Gamma_{Q_k}$ is

$$\Gamma_{Q_k}=\overline{\bigcup_\gamma E_\gamma},$$

where $E_\gamma$ consists of all $t\in\operatorname{Conv}(Q_k)$ such that $\Re\Pi_\gamma(t)=0$ and the corresponding WKB cut system is positive. The union is taken only over cycles occurring in an admissible critical graph of $\omega_t$. The endpoints of the tree occur when an active cycle collapses, and these endpoints are precisely the zeros of $Q_k$.

This conjecture gives a WKB and Stokes-geometric description of the predicted spectral tree. It is motivated by the Bohr–Sommerfeld condition and the expectation that lower-order coefficients affect individual roots only at subleading order, not the limiting carrier. The description remains unproved in general.