The sharp Gaussian decay conjecture for Fourier uncertainty classes

Let $0<a<1$, and let $C_a^2(f)$ and $C_a^2(\widehat f)$ be the weighted Gaussian $L^2$ quantities defined by

$$C_a^2(f)=\left(\int_{\mathbb R}|f(x)|^2e^{2a\pi x^2}\,dx\right)^{1/2},\qquad C_a^2(\widehat f)=\left(\int_{\mathbb R}|\widehat f(x)|^2e^{2a\pi x^2}\,dx\right)^{1/2}.$$

Sharp Gaussian decay conjecture. If both quantities are finite, then

$$fe^{a\pi(\,\cdot\,)^2}\in L^{\infty}(\mathbb R).$$

The preceding theorem establishes the same conclusion with an arbitrary epsilon loss in the exponent, while the Gaussian example motivates removing that loss; the supplied text does not state whether this sharper assertion has been resolved.