according to its orientation. These limiting values satisfy the jump relation $$\label{jumps:M}
    \Psi_{+}(z) = \Psi_{-}(z)J_{\Psi}(z),\qquad z\in \cup_{j=0}^5\Sigma_j,$$ where $$\label{def:JPsi}
    J_{\Psi}(z):= \left\{
            \begin{array}{ll}
              \begin{pmatrix} 0&1&0 \\ -1&0&0 \\ 0&0&1 \end{pmatrix}, & \qquad \hbox{$z\in \Sigma_0$,} \\
              \begin{pmatrix} 1&0&0 \\ 1&1&1 \\ 0&0&1 \end{pmatrix}, & \qquad \hbox{$z\in \Sigma_1$,} \\
              \begin{pmatrix} 1&0&0 \\ 0&1&0 \\ 1&1&1 \end{pmatrix}, & \qquad \hbox{$z\in \Sigma_2$,} \\
              \begin{pmatrix} 0&0&1 \\ 0&1&0 \\ -1&0&0 \end{pmatrix}, & \qquad \hbox{$z\in \Sigma_3$,} \\
              \begin{pmatrix} 1&0&0 \\ 0&1&0 \\ 1&-1&1 \end{pmatrix}, & \qquad \hbox{$z\in \Sigma_4$,} \\
              \begin{pmatrix} 1&0&0 \\ 1&1&-1 \\ 0&0&1 \end{pmatrix}, & \qquad \hbox{$z\in \Sigma_5$.}
            \end{array}
          \right.$$

-   As $z \to \infty$ and $\pm{\mathrm{Im}\,}z>0$, we have $$\label{eq:asyPsi}
     \Psi(z)=
    \sqrt{\frac{2\pi}{3}}i e^{\frac{\rho^2}{6}} \Psi_0
    \left(I+ \frac{\Psi_1}{z} +\mathcal O(z^{-2}) \right)\operatorname{diag}\left(z^{-\frac13},1,z^{\frac13} \right)L_{\pm} e^{\Theta(z)},$$ where $$\label{asyPsi:coeff}
    \Psi_0 =  \begin{pmatrix}
    1 & 0 & 0 \\
    0 & 1  & 0 \\
    \kappa_3(\rho)+\frac{2\rho}{3} & 0 & 1
    \end{pmatrix}, \qquad
    \Psi_1 = \begin{pmatrix}
    0 & \kappa_3(\rho) & 0 \\
    \widetilde\kappa_6(\rho) & 0 & \kappa_3(\rho) + \frac{\rho}{3} \\
    0 & \widehat \kappa_6(\rho)  & 0
    \end{pmatrix},$$ with $$\label{poly:kappa3}
        \kappa_3(\rho) = \frac{\rho^3}{54} - \frac{\rho}{6},$$ $\widetilde\kappa_6(\rho) = \kappa_6(\rho) + \frac{\rho}{3} \kappa_3(\rho) - \frac{1}{3}$, $\widehat\kappa_6(\rho) = \kappa_6(\rho) -\kappa_3^2(\rho) + \frac{\rho^2}{9} - \frac{1}{3},$ and $$\label{poly:kappa6}
        \kappa_6(\rho) = \frac{\rho^6}{5832} -  \frac{\rho^4}{162} - \frac{\rho^2}{72} + \frac{7}{36}.$$ Moreover, $L_{\pm}$ 