Recall that $\tilde{\omega}_{t}=\omega_{0}+t\omega_{M}+\sqrt{-1}\partial\overline{\partial}\varphi_{t}$, so that

$$p^{*}T_{-\sigma}^{*}\tilde{\omega}_{t}=p^{*}\omega_{0}+tp^{*}T_{-\sigma}^{*}\omega_{M}+\sqrt{-1}\partial\overline{\partial}(\varphi_{t}\circ T_{-\sigma}\circ p).$$

We now fix a compact set $K\subset M\backslash S$, which we can assume is sufficiently small so that $f(K)\subset B$ for a ball $B$ as before, and that there is a compact set $K^{\prime}\subset B\times\mathbb{C}^{n-m}$ such that $p:K^{\prime}\to T_{\sigma}(K)$ is a biholomorphism. From (4.16) (together with the $L^{\infty}$ bound for $\varphi_{t}$ from [9, 10]) we see that

$$\|\varphi_{t}\circ T_{-\sigma}\circ p\|_{C^{k}(K^{\prime},\delta)}\leqslant C(k),$$

and therefore also

(4.17)

$$\|\varphi_{t}\|_{C^{k}(K,\omega_{M})}\leqslant C(k),$$

since $T_{-\sigma}\circ p:K^{\prime}\to K$ is a fixed biholomorphism. From [38] we know that $\varphi_{t}\to f^{*}\varphi$ in $C^{1,\alpha}_{loc}(M\backslash S,\omega_{M})$, and so (4.17) implies that $\varphi_{t}\to f^{*}\varphi$ in $C^{\infty}_{loc}(M\backslash S,\omega_{M})$, and therefore that $\tilde{\omega}_{t}\to f^{*}\omega$ in $C^{\infty}_{loc}(M\backslash S,\omega_{M})$. ∎