= Solution
Fix $x\in\mathbb R\setminus\{0\}$ and, before $x$ is swallowed, put $X_t=(g_t(x)-U_t)/\sqrt\kappa$. The <Chordal Loewner equation> and $U_t=\sqrt\kappa B_t$ show, after changing the sign of the Brownian motion, that
$$
dX_t=dW_t+\frac{2/\kappa}{X_t},dt.
$$
Thus $X$ is a <Bessel process> of dimension
$$
\delta=1+\frac4\kappa.
$$
A Bessel process hits zero exactly when $\delta<2$, which here is equivalent to $\kappa>4$. Hitting zero is precisely the swallowing of a nonzero boundary point by the <SLE> hull. For $\kappa\leq4$ the trace is simple and swallows no such point; for $\kappa>4$ swallowing occurs through a boundary contact. Hence the trace intersects $\partial\mathbb H\setminus\{0\}$ exactly when $\kappa>4$.
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