Let $A_n=\{z\in \mathbb{C}: \epsilon_n\leq |z| \leq 1\}$ and $$ f_n\colon A_n\to \mathbb{C}, \quad n\in \mathbb{N}, $$ be a sequence of holomorphic functions such that $\lim_{n\to \infty} \epsilon_n=0$, and for any $r<1$, $\{f_n|_{\{z\in \mathbb{C}: r\leq |z| \leq 1\}}\}_{n\in \mathbb{N}}$ converges to $0$ in $C^\infty$-norm (Uniformly with all derivatives).

Is it true that either $\lim_{n\to \infty} ||f_n(x)||_{\infty} =0$ for all $x$ or (after passing to a subsequence) there are sequences of points $\{z_n\}_{n\in\mathbb{N}}$ and $\{w_n\}_{n\in \mathbb{N}}$ in the boundary components $\{|z|=\epsilon_n\}_{n\in \mathbb{N}}$ with $$ \Big(\lim_{n\to \infty} f_n(z_n) \neq \lim_{n\to \infty} f_n(w_n) \Big) \in \mathbb{C}\cup \{+\infty\} \cup \{-\infty\}. $$