Sharp Holder regularity for autonomous Beltrami equations

Let $\mathrm{\Omega }\subset \mathbb{C}$ be a domain and let $\mathcal{H}:\mathbb{C}\to \mathbb{C}$ be Lipschitz with constant $0<k<1$. We prove that every weak $W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,2}$ solution of the autonomous Beltrami equation $f_{\bar{z}}=\mathcal{H}(f_z)$ belongs to $C_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,\gamma }$, where $\gamma =3\mathrm{/}(2K+1)$ and $K=(1+k)\mathrm{/}(1-k)$. The exponent is optimal, resolving the conjecture of Astala, Clop, Faraco, Jääskeläinen and Koski.