Thanks to Lemma 2.3, we can choose a small $\nu_{0}>0$ so that $\beta_{i}>\nu_{0}\gamma_{i}$ for each $i$ such that $\gamma_{i}>0$. It follows from the description in (2.4) that for any $0<\nu\leqslant\nu_{0}$, the set $\pi^{-1}(E_{\nu})\cap Z$ contains an open neighborhood of the point $q\in D_{\ell}\cap\overline{\pi^{-1}(\mathcal{S}^{c})}$. Repeating this finitely many times on a covering of $\sum_{\ell}D_{\ell}\cap\pi^{-1}(\mathcal{S}^{c})$, and using Lemma 2.2, we find $\nu_{0}>0$ such that for any $0<\nu\leqslant\nu_{0}$, the set $\pi^{-1}(E_{\nu}\cup\mathcal{S})$ contains an open neighborhood of $\tilde{V}+\sum_{\ell}D_{\ell}$. This immediately implies that $E_{\nu}\cup\mathcal{S}$ contains an open neighborhood of $p$, since $\pi$ is an isomorphism away from $\tilde{V}+\sum_{\ell}D_{\ell}$. ∎