The hypoellipticity conjecture for Dirac cohomology

Let $G(\mathbb{R})\supset K(\mathbb{R})$ be a real reductive group, let $\pi$ be an irreducible admissible representation, and let $X_\pi$ be its Harish-Chandra module. A representation is hypoelliptic when its global character is not identically zero on the regular elements of a fundamental Cartan subgroup. Hypoellipticity conjecture.

$$H_D(X_\pi)\neq 0\quad\Longrightarrow\quad \pi\text{ is hypoelliptic}.$$

For unequal-rank groups, hypoellipticity is proposed as the analogue of ellipticity; the source does not provide a proof or counterexample for this implication.