WKB accuracy conjecture for the direct spectral transform

Suppose that $A$ and $S-S_\infty$ are Schwartz-class functions for some constant $S_\infty\in\mathbb{R}$, and that $k\in\mathbb{C}\setminus\{0\}$ admits a global classical solution $f(x,y;k)$ of the eikonal problem. Let $\boldsymbol{\psi}^\epsilon(z;k)$ be the solution of the direct scattering problem at $t=0$. WKB accuracy conjecture. As $\epsilon\downarrow 0$, one has

$$\mathrm{e}^{-f(x,y;k)/\epsilon}\mathrm{e}^{-\mathrm{i} S(x,y)\sigma_3/(2\epsilon)}\boldsymbol{\psi}^\epsilon(x+\mathrm{i} y;k)=\frac{\alpha_0(x,y;k)}{2k}\begin{bmatrix}2\partial f(x,y;k)-\mathrm{i} \partial S(x,y)\\ A(x,y)\end{bmatrix}+o(1),$$

with convergence in a suitable norm, and the $o(1)$ term can be uniquely continued to a full asymptotic power series in positive integer powers of $\epsilon$. The conjecture concerns the accuracy of the leading WKB approximation in the regime where the eikonal problem has a global classical solution; numerical evidence supports it, but a proof of the stated accuracy and full expansion is not supplied here.