We focus on $\gamma_{1}$. The 2-form $\sqrt{-1}d\eta_{2}\wedge(d\bar{\eta}_{2}-d\bar{\eta}_{1})$ on $S$ is exponentially small along the $\mathfrak{D}_{2},\mathfrak{D}_{3}$ ends, so the only divergence problem happens at infinity along the $\mathfrak{D}_{1}$ end.

Applying Lemma 4.1 allows us to replace $\gamma$ by the much simpler function

$$-\frac{1}{4\pi\sqrt{\mathbb{A}}}|(y_{1}-y_{1}^{\prime},y_{2}-y_{2}^{\prime},\mu)|_{a}^{\prime-1}.$$

The integral

$$\int_{S\cap\{y_{2}^{\prime}<\Lambda\}}\gamma(\eta_{1}-\eta_{1}^{\prime},\eta_{2}-\eta_{2}^{\prime},\mu)\sqrt{-1}d\eta_{2}^{\prime}\wedge(d\bar{\eta}_{2}^{\prime}-d\bar{\eta}_{1}^{\prime})$$

has the same divergence behaviour as

$$\int^{\Lambda}-\frac{1}{2\pi\sqrt{\mathbb{A}}}|(y_{1},y_{2}-y_{2}^{\prime},\mu)|_{a}^{\prime-1}dy_{2}^{\prime}\sim-\frac{1}{2\pi\sqrt{Aa_{2\bar{2}}}}\log\Lambda,$$

which is cancelled by the log term we put in the limit. ∎