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We show that the modulus of continuity of the SLE$_4$ uniformizing map isgiven by $(\log \delta^{-1})^{-1/3+o(1)}$ as $\delta \to 0$. As a consequenceof our analysis, we show that the Jones-Smirnov condition for conformalremovability (with quasihyperbolic geodesics) does not hold for SLE$_4$. Wealso show that the modulus of continuity for SLE$_8$ with the capacity timeparameterization is given by $(\log \delta^{-1})^{-1/4+o(1)}$ as $\delta \to0$, proving a conjecture of Alvisio and Lawler.

Regularity of the SLE$_4$ uniformizing map and the SLE$_8$ trace