= Solution
The assertion is false. Here $\mathbb H\setminus S=\{z:\operatorname{Im}z>1\}$. If a normalized map $g_S$ existed, then
$$
h(w)=g_S(w+i)
$$
would be a conformal automorphism of $\mathbb H$. Hence
$$
h(w)=\frac{aw+b}{cw+d}
$$
with real coefficients and $ad-bc=1$. Hydrodynamic normalization forces $h(w)\to\infty$ linearly, so $c=0$ and $h(w)=\alpha w+\beta$ with $\alpha>0$ and $\beta\in\mathbb R$. But
$$
g_S(z)=\alpha(z-i)+\beta,
$$
and $g_S(z)-z\to0$ would require $\alpha=1$ and $\beta=i$, contradicting $\beta\in\mathbb R$.
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