Extremal localization conjecture for the Gaussian STFT

Let $\Delta\subset\mathbb{R}^{2}$ be a measurable set of finite measure, let $A=|\Delta|$, let $\varphi=h_{0}$ be the Gaussian, and let $V_{\varphi}f$ denote the short-time Fourier transform of $f$ with window $\varphi$. Write $D_r$ for the disk of radius $r$ centered at the origin. Gaussian STFT localization conjecture. The quantity

$$\sup_{|\Delta|=A}\sup_{f\in M^{p}}\frac{\|V_{\varphi}f\cdot\chi_{\Delta}\|_{p}^{p}}{\|V_{\varphi}f\|_{p}^{p}}$$

is attained if and only if $\Delta=z+D_{\sqrt{A/\pi}}$ for some $z\in\mathbb{R}^{2}$, up to perturbations of Lebesgue measure zero. This is the joint time-frequency analogue of the Donoho–Stark extremal problem; the authors expect a similar concentration result for $M^{p}$ but state that they were unable to prove it.