The parity conjecture for Dirac cohomology

Let $X$ be an irreducible $(\mathfrak{g},K)$-module, and let $H_D^+(X)$ and $H_D^-(X)$ be the even and odd parts of its Dirac cohomology. Dirac cohomology parity conjecture.

$$\operatorname{Hom}_{\widetilde K}(H_D^+(X),H_D^-(X))=0.$$

A related parity condition can fail for the corresponding $\mathfrak{u}$-cohomology when the infinitesimal character is not regular, whereas the source reports that all known examples still satisfy the Dirac-cohomology parity condition; the general assertion remains open.