Let $\psi\colon\mathbb R\to\mathbb C$, and set $$ F[\psi]:=\pi\ \frac{\displaystyle\int_{\mathbb R} |x|\;|\psi(x)|^2\,\mathrm dx}{\displaystyle\int_{\mathbb R}|\psi(x)|^2\,\mathrm dx}\frac{\displaystyle\int_{\mathbb R} |p|\;|\tilde\psi(p)|^2\,\mathrm dp}{\displaystyle\int_{\mathbb R}|\tilde\psi(p)|^2\,\mathrm dp} $$ with $\tilde\psi$ the Fourier transform of $\psi$. In this physics.SE post we tried to minimise $F[\psi]$ (over some "reasonable" function space, e.g., Schwartz), but we didn't get as far as we would like.

Note that if we were to replace $|x|,|p|\to |x|^2,|p|^2$, we would have $F[\psi]\ge \pi/2$, corresponding to the classical result in Fourier analysis (the Heisenberg uncertainty relation). On the other hand, with the exponents as above, the best we could come up with is a result by user Frédéric Grosshans that $$ F[\psi]\ge \frac{\pi^2}{4e}\sim 0.9077 $$

Numerical analysis seems to suggest that the true bound is actually closer to $0.95$. Has this problem been addressed in the mathematical literature? What is the correct minimum for $F$?