Incomparability conjecture for non-Heisenberg uncertainty principles

Let $\mathcal{S}(\mathbb{R})$ denote the Schwartz space, let $\widehat f$ be the Fourier transform of $f$, and for $2\neq q\in(1,\infty]$ define

$$F_q:\mathcal{S}(\mathbb{R})\setminus\{0\}\longrightarrow\mathbb{R}_{>0},\qquad F_q(f)=\frac{\|f\|_q\|\widehat f\|_q}{\|f\|_2\|\widehat f\|_2}=\frac{\|f\|_q\|\widehat f\|_q}{\|f\|_2^2}.$$

Incomparability conjecture. The image of $F_q$ is all of $\mathbb{R}_{>0}$.

This asserts that, for every $q\neq2$, the corresponding uncertainty principle can be arbitrarily stronger or weaker than the Heisenberg uncertainty principle. The source states that the claim is expected but unproved.