The discretized Furstenburg conjecture

Let $0<\delta\ll1$, let $\Omega$ be a $\delta$-separated set of directions, and for each $\omega\in\Omega$ let $R_\omega$ be a $(\delta,1/2)_2$ set contained in a rectangle of dimensions $\approx1\times\delta$ oriented in direction $\omega$. Let $E$ be a $(\delta,1)_2$ set. Discretized Furstenburg conjecture. There exists an absolute constant $c_3>0$ such that

$$\left|\{(x_0,x_1)\in E\times E:x_1,x_0\in R_\omega\text{ for some }\omega\in\Omega\}\right|\lessapprox\delta^{2+c_3}.$$

The conjecture is a discretized form of the Furstenburg dimension problem, and the source states that a positive answer would imply the continuous Furstenburg problem.