Let $\tau<1$ and $0<\epsilon\ll 1$. Let $f$ be a $T^{2}$-invariant function supported on $\{\text{dist}_{g_{a}}(\cdot,\mathfrak{D}_{1})<C_{2}\}$ inside the model space, with norm $\left\lVert f\right\rVert_{C^{k,\alpha}_{\delta,\tau}}=1$, so that $\left\lVert f\right\rVert_{C^{k,\alpha}_{0,\tau}}\lesssim 1$. Then

$$\begin{cases}\left\lVert G_{\text{Taub}}f\right\rVert_{C^{k+2,\alpha}_{-1+\epsilon,\tau}}\leq C,\quad-1<\tau<1-\epsilon,\\ \left\lVert G_{\text{Taub}}f\right\rVert_{C^{k+2,\alpha}_{0,-2+\epsilon}}\leq C,\quad\tau\leq-1.\end{cases}$$

where the constant only depends on $C_{2},\delta,\epsilon,\tau,k,\alpha$ and the uniform ellipticity bound on $a_{ij}$. In particular if

$$\begin{cases}\text{Either }-1<\tau<1,\quad-3+2\epsilon<\delta\leq 0,\\ \text{or }-2+\epsilon<\tau\leq-1,\quad\delta+\tau>-4+2\epsilon,\end{cases}$$

then $\left\lVert\nabla_{\text{Taub}}^{2}G_{\text{Taub}}f\right\rVert_{C^{k,\alpha}_{\delta,\tau}}\leq C.$