The discretized Kakeya conjecture

Let $\mathbb{T}$ be a set of $\delta$-tubes in $\mathbb{R}^n$ pointing in $\delta$-separated directions, and write $\gtrapprox$ for an inequality up to factors of $\delta^{-\varepsilon}$ for every $\varepsilon>0$.

Discretized Kakeya conjecture. The following two assertions should hold:

(A)

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

(B) If $Y(T)\subset T$ is measurable and $|Y(T)|\geq(\log 1/\delta)^{-1}|T|$ for every $T\in\mathbb{T}$, then

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

Assertion (A) implies the Minkowski-dimension part of the Kakeya set conjecture, while (B) implies the Hausdorff-dimension part. The source presents these as conjectural discretized analogues; no resolution status is supplied.