Sharpness and uniqueness conjecture for the average confidence-width uncertainty constant

Let $x$) and $p$ be the canonical position and momentum operators, and let $E_0$ be the ground-state energy of the self-adjoint operator $|x|+|p|$, namely

$$E_0=\inf_{\lVert\psi\rVert=1}\langle\psi,(|x|+|p|)\psi\rangle.$$

The optimal constant $c^*$ is defined by the uncertainty relation in the paper's equation. Sharpness and uniqueness conjecture. The optimal constant is

$$c^*=E_0^*,$$

and it is attained uniquely, up to dilation, translation, and phase, by the ground state of $|x|+|p|$; numerically, $c^*\approx1.217$. The conjecture would close the numerical gap between the lower bound $\pi/e$ and the upper bound $E_0^2$; establishing it, for example through a Fourier rearrangement inequality controlling $\Delta_a p$ through $f^\sharp$, is identified as the main open problem.