Salamanca-Riba–Vogan convex-hull conjecture for unitarily small lowest K-types

Let $X$ be an irreducible, Hermitian $(\mathfrak{g},K)$-module with a unitarily small lowest $K$-type $\delta$. Let its real infinitesimal character be $\Lambda=(\lambda_a(\delta),\nu)\in\mathfrak{h}^*$. Write $W(\mathfrak{g},\mathfrak{h})$ for the Weyl group and $\rho(\mathfrak{g})$ for the half-sum of positive roots.

Salamanca-Riba–Vogan's convex-hull conjecture. If $X$ is unitary, then $\Lambda$ must lie in

$$\lambda_u(\delta)+\operatorname{conv}\bigl(W(\mathfrak{g},\mathfrak{h})\cdot\rho(\mathfrak{g})\bigr).$$

Otherwise, the Hermitian form of $X$ has opposite signatures on two unitarily small $K$-types $\delta_1$ and $\delta_2$ in $X$.

The source identifies this as Salamanca-Riba and Vogan's 1998 conjecture and states that it implies the preceding unitarity-preservation conjecture. It also states that the paper proves it for $G=U(p,q)$.