Conjecture on the multiplicity of the trivial K-type in Dirac cohomology

Let $G$ be a complex group, let $K$ be its maximal compact subgroup, let $\rho$ be the half sum of the relevant positive roots, and let $E(\rho)$ be the representation with highest weight $\rho$. Write $[\operatorname{Spin}:E(\rho)]$ for the multiplicity of $E(\rho)$ in the spin module.

Multiplicity conjecture. A unitary representation with infinitesimal character $\rho/2$ has Dirac cohomology consisting of the trivial $K$-type with multiplicity $[\operatorname{Spin}:E(\rho)]$ or zero.

The preceding theorem proves this assertion for representations unitarily induced from suitable dominant characters of Levi subgroups, and the paper presents the broader statement as evidence-supported but unresolved.