The Polynomial Wolff axioms conjecture

Let $\mathbb{T}$ be a set of $\delta$-tubes in $\mathbb{R}^n$ satisfying the Polynomial Wolff Axioms: for every semi-algebraic set $S$ and every $\delta\leq\lambda\leq1$, the number of tubes with $|T\cap S|\geq\lambda$ obeys

$$\#\{T\in\mathbb{T}:|T\cap S|\geq\lambda\}\lesssim\lambda^{-n}|S|\delta^{1-n}.$$

Polynomial Wolff axioms conjecture. The following assertions should hold:

(A)

$$\left|\bigcup_{T\in\mathbb{T}}T\right|\gtrapprox\sum_{T\in\mathbb{T}}|T|.$$

(B) If $Y(T)\subset T$ and $|Y(T)|\geq(\log 1/\delta)^{-1}|T|$, then

$$\left|\bigcup_{T\in\mathbb{T}}T\right|\gtrapprox\sum_{T\in\mathbb{T}}|T|.$$

(C)

$$\left\|\sum_{T\in\mathbb{T}}\chi_T\right\|_{\frac{n}{n-1}}\lessapprox1.$$

These assertions ask whether the Polynomial Wolff Axioms suffice for Kakeya-type volume, Hausdorff-dimension, and endpoint overlap bounds. The supplied source gives no resolution status.