More on explicit correspondence between gradient trees in $\mathbb{R}$ and holomorphic convex quadrilaterals in $T^{\ast }\mathbb{R}$

For given smooth functions $(f_1,\dots ,f_n)$ on $M$, Fukaya and Oh showed that the moduli space of pseudoholomorphic disks in $T^{\ast }M$ which are bounded by Lagrangian sections $\{L_i^{\epsilon }=\mathrm{graph}(\epsilon d{f}_i)\}$ is diffeomorphic to the moduli space of gradient trees in $M$ which consist of gradient curves of $\{f_i-f_j\}$. When the image of the pseudoholomorphic disk $w_{\epsilon }$ is a polygon in $\mathbb{C}\simeq T^{\ast }\mathbb{R}$, we can describe $w_{\epsilon }$ by a Schwarz-Christoffel map. In \cite{S25}, we proved that pseudoholomorphic disks $w_{\epsilon }$ converge to the gradient tree in the limit $\epsilon \to +0$ when the image of $w_{\epsilon }$ is a generic convex quadrilateral. In this paper, we show such a convergence for any convex quadrilaterals by studying the non-generic case.