By construction $\log|z_{3}|$ is a function of $\eta_{1},\eta_{2},\mu$ with differential

$$d\log|z_{3}|=V_{(1)}d\mu+\text{Re}(\beta_{13}d\eta_{1}+\beta_{23}d\eta_{2})+\text{Im}(\theta^{\infty}_{13}d\eta_{1}+\theta_{23}^{\infty}d\eta_{2}).$$

In particular the positivity of $V_{(1)}$ in $M^{-}$ means $\log|z_{3}|$ is increasing in $\mu$. Around a given point $P\in S\cap M^{-}$, we first show continuity of $z_{3}$ at $P$. Observe

$$\log|z_{3}|(\eta_{1},\eta_{2},\mu)=\log|z_{3}|(\eta_{1},\eta_{2},A^{-1/4})+\int_{A^{-1/4}}^{\mu}V_{(1)}d\mu.$$

Here $\log|z_{3}|(\eta_{1},\eta_{2},\mu=A^{-1/4})$ is locally $L^{\infty}$ by smoothness of $\log z_{3}$ in $\{\mu>0\}$. Applying Proposition 4.16 and neglecting all locally bounded terms, as $(\eta_{1},\eta_{2},\mu)\to(\eta_{1}(P),\eta_{2}(P),0)$,

$$\log|z_{3}|\sim\int_{A^{-1/4}}^{\mu}\frac{A^{1/2}}{2R}d\mu\sim\frac{1}{2}\log(\frac{A^{1/2}\mu+R}{A^{1/4}})\to-\infty,$$

or equivalently $|z_{3}|\to 0$ as required. The case of $z_{4}$ is completely analogous.

Since $(g^{(1)},\Omega^{(1)})$ is $C^{1,\alpha}$-regular by Proposition 4.19, holomorphicity implies that $z_{3},z_{4}$ are $C^{2,\alpha}$-regular in the local chart of Section 4.4. ∎