A simple observation on Heisenberg-like uncertainty principles

A solution is given to a conjecture proposed recently by Y. Wigderson and A. Wigderson concerning a “Heisenberg-like” uncertainty principle. That conjecture is about the image of the map $ f \mapsto \frac{\|f\|_q\|\hat{f}\|_q}{\|f\|_2\|\hat{f}\|_2}, f\in \mathscr{S}(\mathbb{R})\setminus\{0\} $, where $\mathscr{S}(\mathbb{R}) $ stands for the Schwartz class of functions on the real line. Also, a more general question is answered, where the $L_2$ norm is replaced by the $L_p$ norm in the denominator.