The associated-variety equality conjecture for discretely decomposable restrictions

Let $G$ be a real reductive Lie group with Lie algebra $\mathfrak{g}$, let $G'$ be a reductive subgroup with Lie algebra $\mathfrak{g}'$, and let $X$ be the underlying $(\mathfrak{g},K)$-module of a unitary representation. Denote its associated variety by $\mathcal{V}_{\mathfrak{g}_{\mathbb{C}}}(X)$, and write

$$\operatorname{pr}_{\mathfrak{g}\to\mathfrak{g}'}:\mathfrak{g}_{\mathbb{C}}^*\to(\mathfrak{g}'_{\mathbb{C}})^*$$

for the natural projection dual to $\mathfrak{g}'_{\mathbb{C}}\hookrightarrow\mathfrak{g}_{\mathbb{C}}$. Theorem 5.2.1 gives the inclusion of the projected associated variety into the associated variety of any irreducible $\mathfrak{g}'$-module occurring in $X$.

Associated-variety equality conjecture. The inclusion in that theorem is equality:

$$\operatorname{pr}_{\mathfrak{g}\to\mathfrak{g}'}\bigl(\mathcal{V}_{\mathfrak{g}_{\mathbb{C}}}(X)\bigr)=\mathcal{V}_{\mathfrak{g}'_{\mathbb{C}}}(Y).$$

The claim is presented as plausible after a necessary inclusion obtained from homomorphisms of Harish-Chandra modules. The supplied text gives no resolution status.