The virtually symmetric-type bounded multiplicity conjecture for Zuckerman modules

Let $G\supset G'$ be a reductive symmetric pair, and let $\mathfrak{q}$ be a $\theta$-stable parabolic subalgebra of $\mathfrak{g}_{\mathbb{C}}$. Write $A_{\mathfrak{q}}(\lambda)$ for the associated Zuckerman derived functor module and $\overline{A_{\mathfrak{q}}(\lambda)}$ for its unitarization. The parabolic $\mathfrak{q}$ is of virtually symmetric type when there is a $\theta$-stable parabolic subalgebra $\widetilde{\mathfrak{q}}$ of symmetric type such that

$$\widetilde{L}/L\equiv N_G(\widetilde{\mathfrak{q}})/N_G(\mathfrak{q})$$

is compact.

Virtually symmetric-type bounded multiplicity conjecture. If $\mathfrak{q}$ is of virtually symmetric type, then the restriction

$$\overline{A_{\mathfrak{q}}(\lambda)}|_{G'}$$

has a uniformly bounded multiplicity.

This extends the proposed multiplicity-free criterion from symmetric-type parabolics to virtually symmetric-type parabolics, replacing multiplicity-freeness by uniform boundedness. The supplied text gives no resolution status.