Uniqueness and asymptotics of the Toda differential-equation solutions

Let $M>1/2$, let $k$ be any value, and let $\psi^{(1)}_{[k]}(z,\overline{z};\theta)$ denote the Symanzik-rescaled solution of the differential equations for $\psi^{(1)}$ and $\overline{\psi}^{(1)}$. Let $\mathscr S_k$ be the Stokes sectors, and let $\rho$ and $\phi$ be defined by $z=\rho e^{\mathbbm i\phi}$ and $\overline{z}=\rho e^{-\mathbbm i\phi}$. Uniqueness and asymptotics conjecture. For any value of $k$, $\psi^{(1)}_{[k]}(z,\overline{z};\theta)$ is the unique solution having the following properties: it is an entire function of $(z,\overline{z};\theta)$ on a suitable cover of the punctured complex plane, accounting for the branch point at $(z,\overline{z})=(0,0)$ in $p(z)$ and $\overline p(\overline z)$; and, as $\rho\to\infty$ in the sector

$$(z,\overline{z})\in\bigcup_{j=k}^{r+1+k}\mathscr S_{j-\frac{r+1}{2}},$$

it satisfies

$$\psi^{(1)}_{[k]}\sim\left(e^{\theta+2\pi\mathbbm i\frac{k}{r+1}}z^M\right)^{-\frac r2}e^{-2\frac{\rho^{M+1}}{M+1}\cosh\left[\theta+\mathbbm i(M+1)\left(\phi+2\pi\frac{k}{(r+1)(M+1)}\right)\right]}.$$

This conjecture specifies the distinguished solutions used in the ODE/IM correspondence; the source does not provide a resolution or further evidence for the uniqueness assertion.