Wigderson's range conjecture for the Fourier norm functional

Let $\mathcal{S}(\mathbb{R})$ denote the Schwartz space, let $\widehat{f}$ be the Fourier transform of $f$, and let $\|\cdot\|_p$ denote the $L^p$ norm. For $2\neq p\in(1,\infty]$, define

$$\mathcal{F}_{p,2}^{(1)}:\mathcal{S}(\mathbb{R})\setminus\{0\}\to\mathbb{R}_{>0},\qquad \mathcal{F}_{p,2}^{(1)}(f)=\frac{\|f\|_p\|\widehat{f}\|_p}{\|f\|_2\|\widehat{f}\|_2}=\frac{\|f\|_p\|\widehat{f}\|_p}{\|f\|_2^2}.$$

Wigderson's conjecture. The image of $\mathcal{F}_{p,2}^{(1)}$ is all of $\mathbb{R}_{>0}$.

The conjecture concerns the possible values of a product of Fourier-related $L^p$ norms on the real line. Its status is open and it was stated by the authors as Conjecture 4.13 in the cited work by Wigderson.