We use the Green representation of $\bar{\gamma}_{4}$. The total measure

$$\int_{S}\text{Im}(\sqrt{-1}d\eta_{2}^{\prime}\wedge d\bar{\eta}_{1}^{\prime})\leq C,$$

so the contribution to $\bar{\gamma}_{4}$ from the ball $\{|(\eta_{1}^{\prime},\eta_{2}^{\prime},0)|_{a}\lesssim A^{1/4}\}\subset S$ is bounded by $C\varrho^{-1}$. The contributions from the 3 ends are neglegible unless the point $(y_{1},y_{2},\mu)$ inside the region (4.14) is close to $S$ along some $\mathfrak{D}_{i}$; we focus on the case of $\mathfrak{D}_{1}$. The key fact is the exponential decay of the measure: along $\mathfrak{D}_{1}$ we have

$$\text{Im}(\sqrt{-1}d\eta_{2}^{\prime}\wedge d\bar{\eta}_{1}^{\prime})\leq Ce^{-2\pi y_{2}^{\prime}}\sqrt{-1}d\eta_{2}^{\prime}\wedge d\bar{\eta}_{2}^{\prime}.$$

Thus the contribution from the end $\{y_{2}^{\prime}>1\}\cap S$ is controlled by

$$\begin{split}C\int_{0}^{\infty}e^{-2\pi y_{2}^{\prime}}|(y_{1},y_{2}-y_{2}^{\prime},\mu)|_{a}^{\prime-1}dy_{2}^{\prime}\leq C\varrho^{-1}.\end{split}$$

which implies the estimates on $\bar{\gamma}_{4}$.

For $\bar{\gamma}_{i}-\bar{\bar{\gamma}}_{i}$, the main point is that $S$ approaches its asymptotic cylinder at an exponentially fast rate. The rest of the arguments are similar. ∎