 Solution

For $x\in \mathbb{R}\setminus \{0\}$, the centered Loewner imageis the Boundary-point Bessel flow for SLE, a Bessel process of dimensionWhen $\kappa >4$, 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 $\kappa \in \Lambda$.