The fixed-width Lipschitz Kakeya maximal-function conjecture

Let $v$ be a Lipschitz map from $\mathbb R^2$ to the unit circle. For rectangles $R$ satisfying $|\operatorname V R|\geq\delta|R|$ and $\mathsf w\leq\operatorname W R\leq2\mathsf w$, define

$$\operatorname M_{\delta,v,\mathsf w}f(x)=\sup\frac{\mathbf 1_R(x)}{|R|}\int_R|f(y)|\,dy.$$

Fixed-width Lipschitz Kakeya conjecture. For some $1<p<2$ and some finite $N$, for all $0<\delta<1$, all Lipschitz vector fields $v$, and $0<\mathsf w<\frac1{100\|v\|_{\operatorname{Lip}}}$, the operator $\operatorname M_{\delta,v,\mathsf w}$ maps $L^p(\mathbb R^2)$ to $L^{p,\infty}(\mathbb R^2)$ with norm at most $\lesssim\delta^{-N}$. This fixed-width form is stronger than what is needed for the paper's conditional Hilbert-transform results and is not verified in the source.