Polar duality conjecture for position and momentum ellipsoids

Let measurements of position and momentum be modeled by centered ellipsoids $\Omega_X\subset\mathbb{R}_x^n$ and $\Omega_P\subset\mathbb{R}_p^n$ in the position and momentum spaces, respectively. For a centered set $\Omega_X$, write

$$\Omega_X^o=\{p\in\mathbb{R}_p^n: p\cdot x\leq 1\text{ for every }x\in\Omega_X\}$$

for its polar body.

Polar duality conjecture. The ellipsoids $\Omega_X$ and $\Omega_P$ form a dual pair:

$$\Omega_X^o\subset\Omega_P.$$

This is proposed as a physical formulation of the uncertainty principle that is independent of the particular choice of variances and covariances used to measure uncertainty, and could in principle be justified experimentally. The source does not state whether the conjecture has been proved or disproved.