Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 203 / 4 / c

Stop after a positive initial segment and map it out. By the Conformal Markov property of SLE, the future is a fresh SLE in the remaining domain. Part b says that it hits that domain's boundary away from its current and terminal points. Choosing a bounded crosscut neighborhood whose real-boundary part is shielded from the future forces such a hit to occur on the image of the earlier trace with positive probability. Scale invariance and repeated conditional trials upgrade this to probability one. Thus the trace has self-intersections for $\kappa >4$, in agreement with the Phase classification of the SLE trace.