This question is related to these two.

Let $X\subset \mathbb{C}^{n}$ be a bounded domain. I am interested in the following property: there is a sequence of continuous maps $\varphi_n:\overline{X}\to X$ which are holomorphic on $X$ and such that $\varphi_n(x)\to \varphi(x)$, for every $x \in X$. Every convex domain has this property.

If $n=1$ is this property equivalent to $X$ being a Jordan domain?

If $n>1$ and $X$ is pseudoconvex what are some sufficient conditions on $X$ (other than convexity) so that it has this property?

This question is related to these two.

Let $X\subset \mathbb{C}^{n}$ be a bounded domain. I am interested in the following property: there is a sequence of continuous maps $\varphi_n:\overline{X}\to X$ which are holomorphic on $X$ and such that $\varphi_n(x)\to x$, for every $x \in X$. Every convex domain has this property.

If $n=1$ is this property equivalent to $X$ being a Jordan domain?

If $n>1$ and $X$ is pseudoconvex what are some sufficient conditions on $X$ (other than convexity) so that it has this property?