Multiplicity-free restriction conjecture for derived functor modules of symmetric type

Let $(\mathfrak{g},\mathfrak{g}')$ be a reductive symmetric pair, let $\mathfrak{q}$ be a $\theta$-stable parabolic subalgebra of $\mathfrak{g}_{\mathbb{C}}$, and write $A_{\mathfrak q}(\lambda)$ for the corresponding Zuckerman derived functor module, with $\overline{A_{\mathfrak q}(\lambda)}$ its unitarization. The parabolic subalgebra $\mathfrak{q}$ is of symmetric type when $(\mathfrak{g},\mathfrak{l})$ is a symmetric pair, where $\mathfrak{q}=\mathfrak{l}+\mathfrak{u}$. Multiplicity-free restriction conjecture. If $\mathfrak{q}$ is of symmetric type, then the restriction

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

is multiplicity-free for sufficiently regular $\lambda$. This conjecture proposes a sufficient condition for multiplicity-freeness of restrictions of Zuckerman derived functor modules to reductive symmetric subgroups, motivated by propagation results for visible actions. The conjecture is stated without a resolution in the source.