Consider $\mathbb{C}^{2}$ with coordinates $(x,y)$. We take $V=\{y=0\}\subset\mathbb{C}^{2}$, and consider the plurisubharmonic function $\varphi=\log(|x|^{2})$ on $V$, and $F=\log(|y|^{2})$ on $\mathbb{C}^{2}$. Take local extensions

$$\varphi_{1}=\log(|x-y|^{2}),\qquad\varphi_{2}=\log(|x+y|^{2}),$$

of $\varphi$. We resolve the singularities of $G=\{(x-y)(x+y)y=0\}$ by the blowup of the origin $\pi(r,s)=(r,rs)$. In this case $\{r=0\}=\pi^{-1}(0,0)$, and $\pi^{-1}(G)=\tilde{V}+\sum_{\ell=1}^{3}D_{\ell}$ with

$$\tilde{V}=\{s=0\},D_{1}=\{r=0\},D_{2}=\{s=1\},D_{3}=\{s=-1\}.$$

Then

$$\pi^{*}(|y|^{2})=|r|^{2}|s|^{2},\qquad\pi^{*}(|x-y|^{2})=|r|^{2}|1-s|^{2},\qquad\pi^{*}(|x+y|^{2})=|r|^{2}|1+s|^{2}.$$

In this case, we have

$$\pi^{-1}(\mathcal{S})=\{|r|^{2}|1-s|^{2}<e^{\eta}|r|^{2}|1+s|^{2}\}=\{|1-s|^{2}<e^{\eta}|1+s|^{2}\}.$$

Clearly $\pi^{-1}(\mathcal{S})$ contains an open neighborhood of $\tilde{V}\cup\{s=1\}$. Thus, it suffices to show that the set

$$\pi^{-1}(E_{\nu})=\left\{|r|^{2}|1+s|^{2}\leqslant|r|^{2\nu}|s|^{2\nu}\right\}$$

contains an open neighborhood of $(\{r=0\}\cup\{s=-1\})\cap\{|s|>\varepsilon\}$. This is clear, for any $0<\nu<1$.