Let $\{T_{\alpha}\}$ be a collection of $1\times\cdots\times 1\times N$ tubes, where $N\gg 1$, with maximal $1/N$-separated directions, which all are centered at the origin (i.e. they form a bush). In a note of A. Carbery, the author claims on pg. 4 that for such a collection, "it is easy to see" that the $\{T_{\alpha}\}$ satisfy the conjectured optimal Kakeya maximal bound $$\|\sum_{\alpha}c_{\alpha}\chi_{T_{\alpha}}\|_{L^{\frac{n}{n-1}}(\mathbb{R}^{n})}\lesssim_{n}(\log N)^{\frac{n-1}{n}}N^{\frac{n-1}{n}}(\sum_{\alpha}|c_{\alpha}|^{\frac{n}{n-1}})^{\frac{n-1}{n}}$$ for all scalars $c_{\alpha}$.
I am having difficulty seeing that this is case. I can show this in dimension $2$ by using the fact that $\frac{n}{n-1}=2$ and multiplying out the integrand. For higher dimensions, I tried dyadically decomposing space and using Holder's inequality, but this didn't give me the right bound due to too crude an estimate on the measure of the domain of integration.
Partition the ball $B(0,N)$ into the ball $B(0,1)$ and annuli $A_{j}:=\{j\leq|x|<j+1\}$, for $1\leq j\leq N-1$. Now partition each $A_{j}$ into $\sim j^{n-1}$ caps $\{S_{\beta,j}\}$ of diameter $\sim 1$, and let $e_{\beta,j}$ denote the normalized center of $S_{\beta,j}$. On the ball $B(0,1)$, we use Holder's inequality together with the fact that there are $\sim N^{n-1}$ tubes $T_{\alpha}$ to get
$$\int_{B(0,1)}|\sum_{\alpha}c_{\alpha}\chi_{T_{\alpha}}|^{\frac{n}{n-1}}\leq (\sum_{\alpha}|c_{\alpha}|^{\frac{n}{n-1}})\int_{B(0,1)}(\sum_{\alpha}\chi_{T_{\alpha}})^{\frac{1}{n-1}}\lesssim_{n}N\sum_{\alpha}|c_{\alpha}|^{\frac{n}{n-1}}\tag{1}$$
Fix $1\leq j\leq N$. The tube $T_{\alpha}$ doesn't intersect $S_{\beta,j}$ if $|e_{\alpha}-e_{\beta,j}|>\frac{10}{j}$. Since $\#\{\alpha : |e_{\alpha}-e_{\beta,j}|\leq\frac{10}{j}\}=O((N/j)^{n-1})$, we see that $|\sum_{\alpha}\chi_{T_{\alpha}\cap A_{j}}|\lesssim_{n} (N/j)^{n-1}$. Applying Holder's inequality, we have the bound
$$\sum_{\beta}\int_{S_{\beta,j}}|\sum_{\alpha}c_{\alpha}\chi_{T_{\alpha}}|^{\frac{n}{n-1}}\lesssim_{n}\sum_{\beta}\sum_{\alpha\atop{|e_{\alpha}-e_{\beta,j}|\leq\frac{10}{j}}}|c_{\alpha}|^{\frac{n}{n-1}}(N/j) \tag{2}$$
Now for $j$ and $\alpha$ fixed, $\#\{\beta : |e_{\alpha}-e_{\beta,j}|\leq\frac{10}{j}\}\lesssim_{n}1$. Therefore (2) is
$$\lesssim_{n}\frac{N}{j}\sum_{\alpha}|c_{\alpha}|^{\frac{n}{n-1}}$$
Summing over $j$, we conclude that
$$\|\sum_{\alpha}c_{\alpha}\chi_{T_{\alpha}}\|_{L^{\frac{n}{n-1}}}^{\frac{n}{n-1}}\lesssim_{n}N(\sum_{\alpha}|c_{\alpha}|^{\frac{n}{n-1}})\sum_{j}\frac{1}{j}\lesssim N(\log N)\sum_{\alpha}|c_{\alpha}|^{\frac{n}{n-1}} \tag{3}$$
