On some conjectures of the unitary dual of $U(p,q)$

In this manuscript, we introduce the notion of fundamental cases to study the unitary dual of $U(p,q)$. As applications, we prove of a conjecture of Salamanca-Riba and Vogan stated in 1998, as well as the fundamental parallelepiped (FPP) conjecture of Vogan in 2023 for $U(p,q)$.