This proof is parallel to Proposition 3.5, so will be sketchy. We focus on $\gamma_{4}-\bar{\gamma}_{4}$ as the same arguments work for $\gamma_{i}-\bar{\gamma}_{i}$.

We perform Fourier decomposition in the periodic variables $x_{1},x_{2}$,

$$\gamma_{4}-\bar{\gamma}_{4}=\sum_{(n_{1},n_{2})\in\mathbb{Z}^{2}\setminus\{0\}}h_{n_{1},n_{2}}(y_{1},y_{2},\mu)e^{2\pi in_{1}x_{1}+2\pi in_{2}x_{2}}.$$

The zeroth Fourier mode vanishes by construction. Parseval identity combined with Lemma 4.12 shows

$$\sum_{n_{1},n_{2}}|h_{n_{1},n_{2}}|^{2}=\int_{0}^{1}\int_{0}^{1}|\gamma_{4}-\bar{\gamma}_{4}|^{2}dx_{1}dx_{2}\leq CA^{-1/2}.$$

Over the region (4.14), according to Proposition 4.4 and (4.16)

$$\Delta_{a}(\gamma_{4}-\bar{\gamma}_{4})=0,$$

which translates into the Helmholtz type equations

$$\Delta_{a}^{\prime}h_{n_{1},n_{2}}-4\pi\sqrt{-1}\text{Im}(a^{1\bar{2}})(n_{1}\frac{\partial h_{n_{1},n_{2}}}{\partial y_{2}}-n_{2}\frac{\partial h_{n_{1},n_{2}}}{\partial y_{1}})-4\pi^{2}(a^{p\bar{q}}n_{p}n_{q})h_{n_{1},n_{2}}=0.$$

After the variable substitution

$$\tilde{h}_{n_{1},n_{2}}=h_{n_{1},n_{2}}\exp\left(\frac{2\pi i\text{Im}(a_{1\bar{2}})}{\mathbb{A}}(a_{1\bar{1}}n_{2}y_{1}-\text{Re}(a_{1\bar{2}})n_{1}y_{1}+\text{Re}(a_{1\bar{2}})n_{2}y_{2}-a_{2\bar{2}}n_{1}y_{2})\right),$$

these equations become the Helmholtz equations on $\mathbb{R}^{2}_{y_{1},y_{2}}\times\mathbb{R}_{\mu}$,

$$\Delta_{a}^{\prime}\tilde{h}_{n_{1},n_{2}}=k_{n_{1},n_{2}}^{2}\tilde{h}_{n_{1},n_{2}},$$

where we denote $k_{n_{1},n_{2}}=2\pi(a^{p\bar{q}}n_{p}n_{q})^{1/2}(\frac{A}{\mathbb{A}})^{1/2}$.

The rest of the argument is substantially similar to Proposition 3.5. Observe that $|\text{dist}_{g_{a}^{\prime}}(\cdot,\text{Im}(S))-R|\leq CA^{1/4}$. An upper barrier supersolution to the Helmholtz equation is directly constructed as

$$\begin{split}h_{n_{1},n_{2}}^{\prime}=A^{-1/4}&\{\int_{\mathfrak{D}_{1}}e^{-k_{n_{1},n_{2}}|(y_{1},y_{2}-s,\mu)|_{a}^{\prime}}ds+\int_{\mathfrak{D}_{2}}e^{-k_{n_{1},n_{2}}|(y_{1}-s,y_{2},\mu)|_{a}^{\prime}}ds\\ &+\int_{\mathfrak{D}_{3}}e^{-k_{n_{1},n_{2}}|(y_{1}-s,y_{2}-s,\mu)|_{a}^{\prime}}ds\},\end{split}$$

where $ds=dy_{2},dy_{1},-dy_{1}$ on $\mathfrak{D}_{1},\mathfrak{D}_{2},\mathfrak{D}_{3}$ respectively. Comparing the real and imaginary parts of $\tilde{h}_{n_{1},n_{2}}$ with $C(|n_{1}|+|n_{2}|)h_{n_{1},n_{2}}^{\prime}$ yields the result. ∎