The Dirac-index dichotomy conjecture

Let $G$ be an equal-rank real reductive group with maximal compact subgroup $K$, and let $\widetilde K$ be the spin double cover of $K$. For an irreducible unitary $(\mathfrak{g},K)$ module $\pi$, write $H_D(\pi)=H_D^+(\pi)\oplus H_D^-(\pi)$ for its Dirac cohomology and its even and odd parts. Dirac-index dichotomy conjecture. If $H_D(\pi)\neq 0$ and

$$\operatorname{Hom}_{\widetilde K}(H_D^+(\pi),H_D^-(\pi))\neq 0,$$

then the Dirac index

$$\operatorname{DI}(\pi)=H_D^+(\pi)-H_D^-(\pi)$$

must vanish. This conjecture predicts a dichotomy among the spin lowest $K$-types whenever cancellation occurs. It sharpens a previously stated conjecture that was disproved for the quaternionic real form $F_{4,s}$; the general assertion remains open.