1
\end{pmatrix}, & \hbox{for $z$ in the region $\texttt{V}$,}
       \end{array}
     \right.\end{aligned}$$ where $\widetilde \Psi$ is defined in . Then, $X$ satisfies the following RH problem.

\[rhp:X\] The function $X$ defined in has the following properties:

1.  $X(z)$ is defined and analytic in $\mathbb{C}\setminus \{\cup^5_{j=0}\Sigma_j^{(s)}\cup \{-s\} \cup\{s\}\}$, where $$\label{def:sigmais}
    \begin{aligned}
    &\Sigma_0^{(s)}=(s,+\infty), ~~ &&\Sigma_1^{(s)}=s+e^{\frac{\pi i}{4}}(0,+\infty), ~~ &&&\Sigma_2^{(s)}=-s+e^{\frac{ 3 \pi i}{4}}(0,+\infty),
    \\
    &\Sigma_3^{(s)}=(-\infty, -s), ~~ &&\Sigma_4^{(s)}=-s+e^{-\frac{3\pi i}{4}}(0,+\infty), ~~ &&&\Sigma_5^{(s)}=s+e^{-\frac{\pi i}{4}}(0,+\infty),
    \end{aligned}$$ with the orientations from the left to the right; see the solid lines in Figure \[fig:X\].

2.  $X$ satisfies the jump condition $$\label{eq:X-jump}
     X_+(z)=X_-(z)J_X(z), \qquad z\in \cup^5_{j=0}\Sigma_j^{(s)},$$ where $$\label{def:JX}
    J_X(z):=\left\{
     \begin{array}{ll}
              \begin{pmatrix} 0&1&0 \\ -1&0&0 \\ 0&0&1 \end{pmatrix}, & \qquad \hbox{$z\in \Sigma_0^{(s)}$,} \\
              \begin{pmatrix} 1&0&0 \\ 1&1&1 \\ 0&0&1 \end{pmatrix},  & \qquad  \hbox{$z\in \Sigma_1^{(s)}$,} \\
              \begin{pmatrix} 1&0&0 \\ 0&1&0 \\ 1&1&1 \end{pmatrix},  & \qquad \hbox{$z\in \Sigma_2^{(s)}$,} \\
              \begin{pmatrix} 0&0&1 \\ 0&1&0 \\ -1&0&0 \end{pmatrix}, & \qquad  \hbox{$z\in \Sigma_3^{(s)}$,} \\
              \begin{pmatrix} 1&0&0 \\ 0&1&0 \\ 1&-1&1 \end{pmatrix}, & \qquad  \hbox{$z\in \Sigma_4^{(s)}$,} \\
              \begin{pmatrix} 1&0&0 \\ 1&1&-1 \\ 0&0&1 \end{pmatrix}, & \qquad  \hbox{$z\in \Sigma_5^{(s)}$.}
            \end{array}
          \right.$$

3.  As $z \to \infty$ and $\pm {\mathrm{Im}\,}z>0$, we have $$\label{eq:asyX}
    X(z)=
    \sqrt{\frac{2\pi}{3}}i e^{\frac{\rho^2}{6}} \Psi_0
    \left(I+ \frac{\mathsf{X}_1}{z} +\mathcal O(z^{-2}) \right)\operatorname{diag}\left(z^{-\frac13},1,z^{\frac13} \right)L_{\pm} e^{\Theta(z)},$$ where $\Psi_0$, $L_{\pm}$ and $\Theta(z)$ are 