Discretized classical sticky Kakeya conjecture in $\mathbb{R}^n$

Fix a Hörmander-type phase function $\phi$. For parameters $\varepsilon,\eta,\delta_0>0$, let $\mathrm{SK}(\phi,\varepsilon,\eta,\delta_0)$ denote the assertion that for every $\delta\in(0,\delta_0)$ and every collection of shaded $(\delta,\phi)$-tubes satisfying: the tubes are essentially distinct; for every $\delta\leqslant\rho\leqslant1$ the collection can be covered by $(\rho,\phi)$-tubes, at most $\delta^{-\eta}$ of which are pairwise essentially parallel; and $\sum_{T\in\mathbb{T}}|Y(T)|\geqslant\delta^\eta$, one has

$$\left|\bigcup_{T\in\mathbb{T}}Y(T)\right|\geqslant\delta^\varepsilon.$$

Discretized sticky Kakeya conjecture. For every $\varepsilon>0$ there exist $\eta=\eta_{n,\mathrm{rest}}(\varepsilon)>0$ and $\delta_0=\delta_{0,n,\mathrm{rest}}(\varepsilon)>0$ such that

$$\mathrm{SK}(\phi_{n,\mathrm{rest}},\varepsilon,\eta,\delta_0)$$

holds.

This is the discretized formulation of the classical sticky Kakeya conjecture in dimension $n$, designed to connect sticky Kakeya geometry with restriction estimates. The source provides no evidence of a proof or refutation.