Cohomological-building-block conjecture for integral-character unitary representations of \mathrm{Sp}$p,q$

Let $G_{\mathbb R}=\mathrm{Sp}(p,q)$, and suppose that $\pi$ is a unitary representation of $G_{\mathbb R}$ with integral infinitesimal character. A $\theta$-stable parabolic subalgebra is a decomposition $\mathfrak{q}=\mathfrak{l}\oplus\mathfrak{u}$ stable under the Cartan involution; write $L_{\mathbb R}$ for the normalizer of $\mathfrak{l}$ in $G_{\mathbb R}$. Let $\mathrm{Unip}'(\mathcal{O}_0)$ denote the specified class of unipotent representations attached to a nilpotent orbit $\mathcal{O}_0$, and let $A_{\mathfrak{q}}(\lambda)$ denote cohomologically induced modules.

Cohomological-building-block conjecture. There exist a $\theta$-stable parabolic $\mathfrak{q}=\mathfrak{l}\oplus\mathfrak{u}$ such that

$$L_{\mathbb R}\simeq \mathrm{Sp}(p_0,q_0)\times\mathrm{U}(p_1,q_1)\times\cdots\times\mathrm{U}(p_r,q_r),$$

a representation $\pi_0\in\mathrm{Unip}'(\mathcal{O}_0)$ for some nilpotent orbit of $\mathrm{Sp}(p_0,q_0)$, and weakly fair $A_{\mathfrak{q}}(\lambda)$ modules $\pi_i$ for $\mathrm{U}(p_i,q_i)$, $1\leq i\leq r$, such that $\pi$ is cohomologically induced from

$$\pi_0\boxtimes\pi_1\boxtimes\cdots\boxtimes\pi_r$$

in the weakly fair range.

The conjecture proposes that the unitary dual with integral infinitesimal character is assembled from unipotent representations for smaller symplectic groups and weakly fair cohomologically induced modules for unitary factors. The supplied evidence says that all representations appearing in the conjecture are unitary, but leaves open whether this construction is exhaustive.