We can assume that $K$ is sufficiently small so that $f(K)\subset B$ for a ball $B$ as before, and that there is a compact set $K^{\prime}\subset B\times\mathbb{C}^{n-m}$ so that $p:K^{\prime}\to T_{\sigma}(K)$ is a biholomorphism. We then have

$$\begin{split}\sup_{K}|\mathrm{Sec}(\tilde{\omega}_{t})|&=\sup_{T_{\sigma}(K)}|\mathrm{Sec}(T_{-\sigma}^{*}\tilde{\omega}_{t})|=\sup_{K^{\prime}}|\mathrm{Sec}(p^{*}T_{-\sigma}^{*}\tilde{\omega}_{t})|\\ &=\sup_{\lambda_{t}^{-1}(K^{\prime})}|\mathrm{Sec}(\lambda_{t}^{*}p^{*}T_{-\sigma}^{*}\tilde{\omega}_{t})|.\end{split}$$

For $t>0$ small enough, the sets $\lambda_{t}^{-1}(K)$ are all contained in a fixed compact set $K^{\prime\prime}\subset B\times\mathbb{C}^{n-m}$. From (4.3) and (4.11) we then get a uniform bound for the sectional curvatures of $\lambda_{t}^{*}p^{*}T_{-\sigma}^{*}\tilde{\omega}_{t}$ on $K^{\prime\prime}$, and this proves (4.15). ∎