First we apply estimates similar to those in Lemma 4.2 to the differential $d\psi$, which we think of as an element in $(\mathbb{R}^{d}\oplus\mathbb{R}^{d})\otimes((\mathbb{R}^{d})^{*}\oplus(\mathbb{R}^{d})^{*})$.

Note that the complex structures $J_{Z_{a}}$ and $J_{W_{a}}$ would match exactly via $d\psi$ if there were no $\nabla\mathfrak{X}$ terms (this is what happens in the charts $V_{w}^{\beta^{\vee}}$ where $\mathfrak{X}$ is constant).

According to (2) of Lemma 3.2 $\langle v,\mathfrak{X}\rangle=1$ in $U_{v}^{\beta}$. Hence the projection operator

has a norm of order 1 when restricted to $Z_{a}$, and the desired bound on $d\psi\circ J_{Z_{a}}\circ(d\psi)^{-1}-J_{W_{a}}$ will follow from estimating the $\nabla\mathfrak{X}$ terms.

But according to the Lemma 4.2 we have the uniform bounds:

On the other hand, by (3) of Lemma 3.2 the gradient $\nabla\mathfrak{X}(\log|z|)$ is bounded by $O(\frac{1}{\beta^{\vee}})\cdot\left|g_{\lambda-c,\nu}(\log|z|)\right|$. ∎