Barbasch–Pandžić multiplicity-one conjecture for spin-lowest K-types

Let $G$ be a connected complex simple Lie group, let $\widehat{G}^d$ denote the representations with nonzero Dirac cohomology, and let $V_{\mathfrak{k}}(\eta)$ be an irreducible $K$-type of highest weight $\eta$. A spin-lowest $K$-type is a $K$-type minimizing the spin norm.

Barbasch–Pandžić multiplicity-one conjecture. For every $\pi\in\widehat{G}^d$, there is a unique spin-lowest $K$-type $V_{\mathfrak{k}}(\eta)$, and it occurs in $\pi$ with multiplicity one.

This conjecture makes the proposed description of Dirac series more precise by asserting both uniqueness and multiplicity one for the spin-lowest $K$-type. The paper proves the assertion for complex connected classical Lie groups and the Spin groups; it remains unproved in the full generality stated.