= Solution
Write $D=\mathbb H\setminus K$. By definition of a <compact H-hull>, $D$ is a proper <simply connected domain>, and it agrees with the <complex upper half-plane> outside a sufficiently large disc. The <Riemann mapping theorem> supplies a <conformal isomorphism> $f:D\to\mathbb H$. Choose its boundary normalization so that the <prime end> at infinity maps to infinity. This <prime end> is unambiguous because the domain has an ordinary straight real boundary near infinity.
The <Schwarz reflection principle>, applied there in the coordinate $1/z$, extends $f$ across the real boundary near infinity. It has a simple pole there, with <Laurent series>
$$
f(z)=az+b+\frac{c}{z}+O(z^{-2}),\qquad a>0,\quad b,c\in\mathbb R.
$$
The pole is simple because the reflected map is locally conformal at infinity; its leading coefficient is positive because it maps the upper side to the upper side. Postcomposing with the half-plane automorphism $w\mapsto(w-b)/a$ gives
$$
g_K(z)=z+O(1/z),\qquad\boxed{g_K(z)-z\longrightarrow0.}
$$
This is <hydrodynamic normalization at infinity>.
If $g_1,g_2$ both satisfy it, $g_2\circ g_1^{-1}$ is a <conformal automorphism of the upper half-plane>. The asymptotics force it to fix infinity, so it has the form $w\mapsto\alpha w+\beta$ with $\alpha>0$ and $\beta\in\mathbb R$. The same asymptotics force $\alpha=1$, $\beta=0$. Hence \b[the normalized <mapping-out function> exists and is unique]. The argument requires reflection only outside a large disc, not regularity of the hull's entire boundary.
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