A uniform Hilbert transform estimate for Lipschitz directions

We prove a uniform L${}^2$ bound for the Hilbert transform along a Lipschitz unit vector field in the plane, with integration restricted to a fixed absolute multiple of the reciprocal Lipschitz seminorm. The bound is uniform in the inner truncation and holds for fields depending on both coordinates. This gives an affirmative answer to Stein's weak-$(2,2)$ conjecture.