Wigderson–Wigderson conjecture on the range of Fourier norm ratios

Let $\mathcal{S}(\mathbb{R})$ be the space of Schwartz functions, and let $\hat f$ denote the Fourier transform of $f$. For a non-zero $f\in\mathcal{S}(\mathbb{R})$ and $q\in(1,\infty]$, define

$$F_q(f)=\frac{\|f\|_q\|\hat f\|_q}{\|f\|_2\|\hat f\|_2}=\frac{\|f\|_q\|\hat f\|_q}{\|f\|_2^2}.$$

Wigderson–Wigderson conjecture. For all $q\neq 2$, the image of

$$F_q:\mathcal{S}(\mathbb{R})\setminus\{0\}\to\mathbb{R}_{>0}$$

is $\mathbb{R}_{>0}$.

This conjecture asks whether the norm ratio arising in the generalized Heisenberg uncertainty principle can attain every positive real value when $q\neq 2$. The supplied source does not state whether the conjecture has been resolved.