Two-ends Furstenberg conjecture

Let $\delta\in(0,1)$, and let $(L,Y)_\delta$ be a set of directionally $\delta$-separated lines in $\mathbb{R}^n$ with an $(\varepsilon_1,\varepsilon_2)$ two-ends, $\lambda$-dense shading. Write $E_L:=\bigcup_{\ell\in L}Y(\ell)$. Two-ends Furstenberg conjecture. For every $\varepsilon>0$,

$$|E_L|\geq c_\varepsilon\delta^\varepsilon\delta^{O(\varepsilon_1)}\lambda^{\frac{n-1}{2}}\sum_{\ell\in L}|Y(\ell)|.$$

The paper proposes this estimate as a two-ends Furstenberg inequality implying the restriction conjecture, and proves it in the plane using a Furstenberg set estimate; the general higher-dimensional statement remains open.