Rearrangement conjecture for characteristic functions and quadratic spectral concentration

Let $A\subset\mathbb{R}$ be measurable with finite measure, and let $\chi_A$ be its characteristic function. Write $A^*$ for the centered interval with $|A^*|=|A|$, so that $\chi_A^*$ is the symmetric decreasing rearrangement of $\chi_A$. For $W>0$, compare the quadratic spectral concentrations

$$\int_{-W/2}^{W/2}|\widehat{\chi_A}(\xi)|^2\,d\xi\quad\text{and}\quad\int_{-W/2}^{W/2}|\widehat{\chi_A^*}(\xi)|^2\,d\xi.$$

Characteristic-function rearrangement conjecture. For every measurable $A$ with $|A|<\infty$ and every $W>0$,

$$\int_{-W/2}^{W/2}|\widehat{\chi_A}(\xi)|^2\,d\xi\leq\int_{-W/2}^{W/2}|\widehat{\chi_A^*}(\xi)|^2\,d\xi.$$

This is the proposed restriction of the broader Donoho–Stark rearrangement principle to functions taking only the values $0$ and $1$. The general rearrangement inequality is known to fail beyond the low-frequency range, whereas this characteristic-function version is presented as an open conjecture and is related to weighted $L^2$ inequalities for non-harmonic trigonometric polynomials.