Stein's Hilbert transform conjecture for Lipschitz vector fields

Let $u:\mathbb{R}^2\to\mathbb{R}$ be a Lipschitz vector field, let $\gamma(x,y,t)=u(x,y)t$, and choose $\epsilon_0>0$ as above. Define the Hilbert transform along the corresponding variable curve by

$$H_{\Gamma}f(x,y)=\operatorname{p.v.}\int_{-\epsilon_0}^{\epsilon_0}f(x-t,y-u(x,y)t)\,\frac{dt}{t}.$$

Stein's conjecture. The operator $H_{\Gamma}$ is bounded on $L^p(\mathbb{R}^2)$ for every $1<p<\infty$.

This is the singular-integral analogue of the preceding differentiation conjecture for Lipschitz vector fields. The source presents the assertion as a conjectural statement and supplies no resolution evidence.