Automorphic isolation conjecture for cohomological representations

Let $G$ be a semisimple algebraic group over ${\mathbb Q}$, let $\mathfrak{q}$ be a $\theta$-stable parabolic subalgebra, and let $L\subset G$ be its associated Levi subgroup. Let $\pi$ be the cohomological representation associated with $\mathfrak{q}$, and write $\widehat{G}_{\rm Aut}$ for the automorphic dual of $G$. Automorphic isolation conjecture. The representation $\pi$ is isolated in

$$\{\pi\}\cup\widehat{G}_{\rm Aut}$$

whenever $L$ has compact center. In particular, this should always hold when ${\rm rank}_{\mathbb C}(G)={\rm rank}_{\mathbb C}(K)$, equivalently when $G$ has discrete series. This conjecture is presented as a conjectural strengthening of isolation in the full unitary dual and is used to derive automorphic and cohomological consequences; its general status is not resolved in the source.