Kakeya $L^p$ conjecture

Fix $d\geq2$. For each $1/R$-cap $\theta$ on $S^{d-1}$, let $\theta^*$ be the corresponding radius-one, length-$R$ tube, and let $x_\theta$ be arbitrary. Kakeya $L^p$ conjecture. For every $\epsilon>0$, there is a constant $C(\epsilon,d)$ such that for every $p\geq1$,

$$\int\left|\sum_\theta1_{\theta^*+x_\theta}(x)\right|^p\,dx\leq C(\epsilon,d)R^\epsilon\left(R^{(d-1)p}+R^d\right).$$

The right-hand side is the benchmark supplied by the concentric-tube example. The source gives no resolution.