= Solution
For $x\in\mathbb R\setminus\{0\}$, the centered Loewner image
$$
Y_t=\frac{g_t(x)-U_t}{\sqrt\kappa}
$$
is the <Boundary-point Bessel flow for SLE>, a Bessel process of dimension
$$
d=1+\frac4\kappa.
$$
When $\kappa>4$, one has $d<2$, and the <Hitting-zero classification for a Bessel process> says that $Y$ hits zero almost surely. Thus every fixed nonzero boundary point is swallowed in finite time.
A hull generated by a continuous trace cannot swallow a real point while the trace remains strictly inside $\mathbb H$: before any contact with the real line, the trace is a crosscut-free interior curve and no boundary interval is disconnected from infinity. Consequently finite swallowing forces the trace to meet $\mathbb R$ away from its initial point. SLE therefore almost surely intersects the boundary for every $\kappa\in\Lambda$.
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