Let $\{\Omega_{j}\}_{j\in\mathbb{N}}$ be a sequence of smooth bounded domains in $\mathbb{C}^{n}$ such that $\Omega_{j}$ converges to a smooth bounded domain $\Omega$ in the sense that the defining function of $\Omega_{j}$, say $\rho_{j}$ converges to the defining function of $\Omega~~$,say $\rho$ uniformly on each compact subset of $\mathbb{C}^{n}$, where a defining function of a smooth bounded domain $D$ is defined as a smooth real-valued function say $\psi$ on $\mathbb{C}^{n}$ such that $\psi$ satisfies the following: $$ \begin{align} D =\{z\in\mathbb{C}^{n}:\psi(z)<0\}&,\\ \partial D =\{z\in\mathbb{C}^{n}:\psi(z)=0\}&,\text{ and}\\ \operatorname{grad}(\rho)\neq0\text{ on }\partial D.& \end{align}$$

This notion of convergence implies the Hausdorff convergence of open domains. Precisely if I say: For any compact set $K\subset\Omega$, then $K\subset\Omega_{j}$ for sufficiently large $j$. Also, for any compact set $L\subset\mathbb{C}^{n}\setminus\overline{\Omega}$, then $L\subset\mathbb{C}^{n}\setminus\overline{\Omega_{j}}$. My question is: for fixed $x_{0}\in \Omega$ can we imply that there exists a positive constant $c$ such that $\min\{P_{j}(x_{0},y):y\in\partial\Omega_{j}\}\geq c$ for large $j$, where $P_{j}$ denotes the Poisson kernel of $\Omega_{j}$. It would be a great help if any suggestions I get.