We know that the HUP inequality is saturated when the wavefunction is Gaussian. In the case of the harmonic oscillator, this is the case for the ground-state. For a general Hamiltonian, and among all its eigenstates, is there a way to see which one of them has the least $\Delta x \Delta p$? Obviously, the lower bound will be saturated iff the wavefunction is Gaussian, but that's not what I'm asking. Is it true that $\Delta x \Delta p$ is always less for the ground-state than its value for other energy eigenstates?
EDIT: as pointed out in the comments, this is trivially false if we don't put any constraints on the form on the Hamiltonian. For the sake of convenience, let us assume the Hamiltonian is of the form $$H = \frac{p^2}{2m} + V(x).$$
