We consider the bi-PIKAS family in a neighborhood of $(\lambda_{0},\nu_{0})$ and extend it by rescaling to (a neighborhood of) the ray $(\lambda,\nu)=\left(\log|a|+\log|s|\cdot\lambda_{0},\frac{\nu_{0}}{\log|s|}\right)$ in $\operatorname{SC}(S)\times\operatorname{SC}(T)$.

The $C_{i}$-estimates for the $(\lambda,\nu)$ bi-PIKAS considered in the bi-PIKAS metric (rather than in Euclidean) are equivalent to the corresponding estimates for the rescaled structure $((\log|s|)^{-1}\lambda,\log|s|\nu)$ made in the Euclidean metric as before. This is because the Euclidean metric on $\partial\Delta_{\lambda_{0}}$ is equivalent to the bi-PIKAS metric on $\Sigma$. But to pass from $(\lambda,\nu)$ to $((\log|s|)^{-1}\lambda,\log|s|\nu)$ we will need to rescale all the parameters as well:

Or equivalently, we could apply the log map with the base $|s|$ as in [HZ02].

We saw in the proof of Lemma 4.4 that $\operatorname{Hess}\Phi^{\mathrm{sm}}_{\lambda_{0}+\gamma_{0}}$ is degenerate along $\mathcal{F}$ outside $\Delta^{\vee}_{\lambda_{0}+\gamma_{0}+h_{0}}$. This argument extended to the entire toric variety shows that up to terms of order $O(\epsilon)$, the set $X_{T}\backslash\log^{-1}(\Delta^{\vee}_{\lambda_{0}+\gamma_{0}+h_{0}})$ (which contains $Z_{s}$) has distance from $\partial\Delta^{\vee}_{\lambda}$ bounded by the diameters of the torus fibers $\mathbb{T}$. The size of the tori $\mathbb{T}$ is determined by the norm of $\operatorname{Hess}\Phi^{\mathrm{sm}}_{\lambda+\gamma}$ at the corresponding point, which is bounded by $\frac{C_{1}(\gamma_{0})}{(\log|s|)^{2}}$.

The rest of the proof consists of careful picks for asymptotics of the rescaled parameters $\beta_{0},\beta^{\vee}_{0},h,\gamma_{0},c_{0},\epsilon$ to ensure that the following expressions

go to 0 as $\log|s|\to\infty$, where the first two lines take care of the Hausdorff convergence, and the last two give matching of the complex structure and the metric under the embedding $\psi:Z^{\mathrm{sm}}_{s}\to W_{s}$

For instance, we can choose

And $\beta_{0}(\log|s|)$ has satisfy $C_{1}(\frac{1}{\log|s|}e^{-\beta_{0}\log|s|})<\log|s|$ (i.e. $e^{-\beta}<\gamma$, which is needed for the proof of Lemma 4.4) and

which is possible due to the fast decreasing factor of $e^{-\beta_{0}\log|s|}$ when $\beta_{0}(\log|s|)$ is changing slowly.

Finally, notice that the bi-PIKAS of type $((\log|s|)^{-1}\cdot\lambda,\log|s|\cdot\nu)=(\lambda_{0}+(\log|s|)^{-1}\log|a|,\nu_{0})$ converges to the $(\lambda_{0},\nu_{0})$-bi-PIKAS. ∎