Guth–Zahl polynomial Wolff axiom maximal inequality conjecture

Let $\mathbb{T}$ be a family of $\delta$-tubes in $\mathbb{R}^n$. For $D\in\mathbb{N}$ and $N\geqslant1$, say that $\mathbb{T}$ satisfies the $(D,N)$-polynomial Wolff axiom if, for every $\lambda\geqslant\delta$ and every semialgebraic set $E\subseteq\mathbb{R}^n$ of complexity at most $D$,

$$\#\{T\in\mathbb{T}:|T\cap E|\geqslant\lambda|T|\}\leqslant N\delta^{-(n-1)}\lambda^{-n}|E|.$$

Guth–Zahl conjecture. Let $p\geqslant\frac{n}{n-1}$. For every $\varepsilon>0$, there are a complexity $D=D_{\varepsilon,n}\in\mathbb{N}$ and a constant $C_{\varepsilon,n}>0$ such that, whenever $0<\delta<1$, $N\geqslant1$, and $\mathbb{T}$ satisfies the $(D,N)$-polynomial Wolff axiom,

$$\Big\|\sum_{T\in\mathbb{T}}\chi_T\Big\|_{L^p(\mathbb{R}^n)}\leqslant C_{\varepsilon,n}N^{1-1/p}\delta^{-(n-1-n/p)-\varepsilon}\Big(\sum_{T\in\mathbb{T}}|T|\Big)^{1/p}.$$

This conjecture, attributed in the source to Guth and Zahl, is described as stronger than the Kakeya maximal conjecture in the relevant direction. The source notes that its precise formulation differs from the cited Guth–Zahl version in some respects, and does not report a resolution.