= Solution
Write $R_j$ for the rectangles, $x_j$ for their centers and $e_j$ for their long-axis directions. For the finite exponent $p$ in the displayed estimate, take smooth rotated cap functions $\psi_j$, with $0\le\psi_j\le1$, equal to one on angular distance at most $\delta/C$ from $e_j$ and supported within $2\delta/C$. Choose $C$ sufficiently large once and for all. The direction separation makes these cap supports disjoint, and $\int|\psi_j|^p\,d\sigma\lesssim\delta$.
Define
$$
f_j(\omega)=e^{ix_j\cdot\omega}\psi_j(\omega),\qquad
h_j(x)=\widehat{f_j\,d\sigma}(x)
=\widehat{\psi_j\,d\sigma}(x-x_j).
$$
The <Fourier modulation and translation identity> gives the second equality. Rotating the <circle cap Fourier lower bound> then gives \b[$|h_j(x)|\gtrsim\delta$ on $R_j$]. The half-side lengths of $R_j$ are no larger than the two frequency bounds used in part (b).
Let $\varepsilon_j$ be independent <Rademacher random variables>. Because the input cap supports are disjoint, for every choice of signs
$$
\left\|\sum_j\varepsilon_j f_j\right\|_{L^p(\sigma)}^p
=\sum_j\|\psi_j\|_{L^p(\sigma)}^p\lesssim M\delta,
$$
where $M=\#\mathcal R$. Apply the assumed <Fourier extension estimate> to the sum. Average over signs and use the <Khintchine inequality> pointwise, followed by the <Tonelli theorem>:
$$
\int_{\mathbb R^2}\left(\sum_j|h_j(x)|^2\right)^{p/2}\,dx
\lesssim_p\mathbb E\left\|\sum_j\varepsilon_jh_j\right\|_p^p
\lesssim_p M\delta.
$$
There is no requirement that the spatial rectangles be disjoint; disjointness is used only for the input caps on the <unit circle>. Their spatial overlaps are precisely what the square function measures. The cap lower bounds now imply
$$
\delta^p\int\left(\sum_j\mathbf1_{R_j}\right)^{p/2}
\lesssim_p M\delta.
$$
Since each rectangle has area $\delta^{-3}$, the <restriction-to-rectangle overlap principle> gives
$$
\boxed{\int\left(\sum_{R\in\mathcal R}\mathbf1_R\right)^{p/2}
\lesssim_p M\delta^{1-p}
=\delta^{4-p}\sum_{R\in\mathcal R}|R|.}
$$
The constants are independent of $\delta$, the centers and the collection. The finite-$p$ interpretation is the one for which the printed power integral is defined. A single cap also shows that the assumed diagonal <Fourier extension estimate> can hold only for $p\ge4$: its output contributes at least $\delta^{p-3}$ to the $p$th-power <norm>, whereas its input contributes at most a constant times $\delta$.
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