WKB asymptotic conjecture for Stokes matrices

Let $A$ be a generic matrix, let $u$ be the diagonal irregular-type parameter, and let $S_+(u,A,\varepsilon)$ be the upper triangular Stokes matrix with $S_-=S_+^\dagger$. Write $\Delta^{(k)}_i$ for the relevant minor coordinates and $\lambda^{(k)}_i$ for the eigenvalues of $S=S_-S_+$. WKB asymptotic conjecture. As $\varepsilon\to0$, the minor coordinates admit an expansion

$$\log\Delta^{(k)}_i(S_+(u,A,\varepsilon))\sim\varepsilon^{-1}\delta^{(k)}_i(u,A)+(\delta^{(k)}_i)_0+\varepsilon(\delta^{(k)}_i)_1+\cdots,$$

and the eigenvalues admit an expansion

$$\log\lambda^{(k)}_i(S(u,A,\varepsilon))\sim\varepsilon^{-1}\eta^{(k)}_i(u,A)+\cdots.$$

The eigenvalue expansion is uniform as $u$ approaches the caterpillar line, $u\to u_{\rm cat}(t)$. This assumption is used to derive rhombus and interlacing inequalities for leading WKB exponents; it is a central unproved input of the paper.