Vogan's fundamental parallelepiped conjecture for real reductive Lie groups

Let $G$ be a real reductive Lie group in the Harish-Chandra class. Let $\pi$ be any fully supported irreducible $(\mathfrak{g}, K)$-module with real infinitesimal character $\Lambda$.

Vogan's FPP conjecture. If $\langle \Lambda, \beta^{\vee}\rangle >1$ for any simple root $\beta$, then $\pi$ is non-unitary.

This conjecture gives a reduction in the classification of irreducible unitary representations by restricting attention to modules whose real infinitesimal characters lie inside the fundamental parallelepiped. The paper's abstract states that the conjecture is proved for complex simple Lie groups, while the formulation here is for real reductive groups in the Harish-Chandra class.