Conformal non-removability of the Brownian graph

We prove that the graph of a standard one-dimensional Brownian motion, ${\mathrm{\Gamma }}_B=\{(t,B_t):t\in [0,1]\}$, is almost surely not conformally removable. This resolves the question left open by Doherty and Miller in their study of the Sobolev removability of the Brownian graph~\cite{DM26}. We also construct non-removable $\alpha$-Hölder graphs for every $\alpha \in (0,2\mathrm{/}3)$, including the previously open range $\alpha \in [1\mathrm{/}2,2\mathrm{/}3)$. Together with the known removability result for $\alpha >2\mathrm{/}3$, this identifies $2\mathrm{/}3$ as the threshold for conformal removability of Hölder graphs.