Conjecture I on tameness of infinitely punctured planes

Let $\mathscr{P}$ denote the class of closed discrete subsets of $\mathbb{C}$, and let $A\in\mathscr{P}$. An automorphism $h\in\operatorname{Aut}(\mathbb{C}\setminus A)$ has infinite order if no positive power of $h$ is the identity. The quotient $\left(\mathbb{C}\setminus A\right)/\langle h\rangle$ is formed by the cyclic group generated by $h$.

Conjecture I. If there exists an automorphism of infinite order $h\in\operatorname{Aut}(\mathbb{C}\setminus A)$ such that the quotient space $\left(\mathbb{C}\setminus A\right)/\langle h\rangle$ has infinitely many punctures, then $A$ is not tame.

The conjecture is motivated by the preceding theorem, which shows that tameness would force arbitrarily many points of $A$ to occur in every prescribed small disk under the same quotient hypothesis. The source provides no resolution, so the conjecture remains open.