Space-filling SLE above parameter eight

For $\kappa >8$, the Loewner trace visits every point of the complex upper half-plane almost surely. Choose $\rho =\kappa -8$ in the SLE interior-point martingale to obtain $M_t={{\rm Y}}_t^{(\kappa -8)/8}{S}_t^{-(\kappa -8)/\kappa }$. The maximal inequality for a nonnegative supermartingale bounds its running supremum. If a fixed interior ball were avoided, its center would have ${{\rm Y}}_t\ge c_1>0$ before swallowing, forcing $S_t\ge c_2>0$. But the Chordal Loewner equation gives $d{Y}_t^2/dt=-4S_t^2$, forcing a finite lifetime with $Y_t\to 0$, and $d\log {{\rm Y}}_t/d\log Y_t=2S_t^2$ then forces ${{\rm Y}}_t\to 0$, a contradiction. Countably many rational balls give a dense trace; continuity and Transience of chordal SLE make its image relatively closed, so it is all of the complex upper half-plane. This argument does not cover the critical parameter eight.