The Lipschitz Kakeya maximal-function conjecture

Let $v$ be a Lipschitz map from $\mathbb R^2$ to the unit circle. For a rectangle $R$ of sufficiently short length, let $\operatorname V R=R\cap v^{-1}(\operatorname{EX}R)$, where $\operatorname{EX}R$ denotes the interval of directions determined by $R$, and define

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

Lipschitz Kakeya maximal-function conjecture. For some $1<p<2$ and some finite $N$, for all $0<\delta<1$ and all Lipschitz vector fields $v$, the operator $\operatorname M_{\delta,v}$ maps $L^p(\mathbb R^2)$ to $L^{p,\infty}(\mathbb R^2)$ with norm at most $\lesssim\delta^{-N}$. The known $L^2$ bound has norm $\lesssim\delta^{-1/2}$, but the asserted subquadratic estimate is not verified in the source.