= Solution
For a unit direction $e$, let $T_\delta(a,e)$ be a length-one <Kakeya tube> with transverse radius $\delta$, centered at $a$, and define the <Kakeya maximal function> by
$$
\mathcal K_\delta f(e)=\sup_a\frac1{|T_\delta(a,e)|}\int_{T_\delta(a,e)}|f(x)|\,dx.
$$
Using normalized surface measure on the direction sphere, the <Kakeya maximal conjecture> is the following family of estimates, in the formulation relevant to this paper:
$$
\boxed{\|\mathcal K_\delta f\|_{L^n(S^{n-1})}
\le C_{n,\varepsilon}\delta^{-\varepsilon}\|f\|_{L^n(\mathbb R^n)}
\quad(0<\delta<1,\ \varepsilon>0).}
$$
The constant is independent of $\delta$ and $f$. Replacing round <Kakeya tubes> by comparable rectangular tubes changes only dimensional constants.
A bounded <Kakeya set> contains a unit line segment in every direction. The <Kakeya Minkowski dimension conjecture> says that every such set has full <Minkowski dimension> $n$. The maximal estimate in fact gives full lower as well as upper <Minkowski dimension>.
To prove that implication, let $E_\delta=\{x:\operatorname{dist}(x,E)<\delta\}$. Each unit segment in $E$ has a thinner tube contained in $E_\delta$, so $\mathcal K_{c\delta}\mathbf1_{E_\delta}(e)\ge1$ for every $e$, with a fixed dimensional $c>0$. Apply the maximal estimate to this <indicator function>:
$$
1\lesssim C_{n,\varepsilon}\delta^{-\varepsilon}|E_\delta|^{1/n},
\qquad |E_\delta|\gtrsim_{n,\varepsilon}\delta^{n\varepsilon}.
$$
If $N_\delta(E)$ is the smallest number of radius-$\delta$ balls covering $E$, that cover, enlarged by a fixed factor, covers $E_\delta$. Thus $|E_\delta|\lesssim_n\delta^n N_\delta(E)$ and
$$
N_\delta(E)\gtrsim_{n,\varepsilon}\delta^{-n+n\varepsilon}.
$$
Taking the lower limit of $\log N_\delta(E)/\log(1/\delta)$ and then letting $\varepsilon\downarrow0$ gives lower <Minkowski dimension> at least $n$. Bounded subsets of $\mathbb R^n$ have upper <Minkowski dimension> at most $n$. Consequently \b[both dimensions equal $n$], which proves the requested implication.
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