By the smooth convergence of $\tilde{\omega}_{k}$ and Lemma 3.2,

$$\lim_{k\longrightarrow\infty}\int_{F_{r,k}(A)}\tilde{\omega}_{k}=\int_{A}\omega_{0}=0,$$

for any cycle $A\in H_{2}(S,\mathbb{Z})$. For $k\gg 1$, we have

$$|\int_{F_{r,k}(A)}\omega_{k}|=i^{2}_{g_{k}}(p_{k})|\int_{F_{r,k}(A)}\tilde{\omega}_{k}|<\frac{1}{2k^{2}}<1.$$

Since $[F_{r,k}(A)]\in H_{2}(M_{k},\mathbb{Z})$ and $[\omega_{k}]\in H^{2}(M_{k},\mathbb{Z})$, we obtain $\int_{F_{r,k}(A)}\omega_{k}\in\mathbb{Z}$. This implies that $|\int_{F_{r,k}(A)}\omega_{k}|=0$, and we obtain the conclusion

$$\int_{A}F_{r,k}^{*}\tilde{\omega}_{k}=0.$$

∎