The Kakeya conjecture for polynomial Wolff axioms

A Kakeya set in $\mathbb{R}^n$ is a compact set containing a unit line segment in every direction. A collection $\mathbb{T}$ of $\delta$-tubes satisfies the polynomial Wolff axioms if it obeys the polynomial incidence bounds for semi-algebraic sets of bounded complexity used in the paper. Write $\chi_T$ for the indicator of $T$, and define $\lessapprox$ by: for every $\epsilon>0$, there is a constant $C_\epsilon$ independent of $\delta$ such that $A\leq C_\epsilon\delta^{-\epsilon}B$.

Kakeya conjecture for the polynomial Wolff axioms. For every dimension $n$, there is a complexity $E$ such that, whenever $\mathbb{T}$ is a set of $\delta$-tubes in $\mathbb{R}^n$ obeying the polynomial Wolff axioms for semi-algebraic sets of complexity at most $E$,

$$\Big\|\sum_{T\in\mathbb{T}}\chi_T\Big\|_{n/(n-1)}\lessapprox 1.$$

This would imply

$$\Big|\bigcup_{T\in\mathbb{T}}T\Big|\gtrapprox 1.$$

The conjecture would extend the known Wolff-axiom maximal-function estimate and would imply the expected near-unit volume bound for unions of tubes satisfying the polynomial axioms; it remains open.