Vogan's fundamental parallelepiped conjecture for real reductive groups

Let $X$ be an irreducible $(\mathfrak{g},K)$-module with real infinitesimal character $\Lambda\in\mathfrak{h}^*$. Assume that $X$ is not cohomologically induced in the good range from any $(\mathfrak{l},L\cap K)$-module $X_L$, for any proper $\theta$-stable parabolic subalgebra

$$\mathfrak{q}=\mathfrak{l}+\mathfrak{u}.$$

For each simple root $\alpha\in\Delta^+(\mathfrak{g},\mathfrak{h})$, let $\alpha^\vee$ denote its coroot.

Vogan's fundamental parallelepiped conjecture. If $X$ is unitary, then

$$\langle\Lambda,\alpha^\vee\rangle\leq 1$$

for every simple root $\alpha\in\Delta^+(\mathfrak{g},\mathfrak{h})$.

This conjecture proposes a reduction of the unitary-dual problem by bounding the real infinitesimal character of representations that are not obtained by good-range cohomological induction. The abstract states that the paper proves Vogan's 2023 conjecture for $U(p,q)$.