Kobayashi's associated-variety conjecture for discretely decomposable restrictions

Let $G$ be a real reductive group, let $G'$ be a reductive subgroup, and choose a maximal compact subgroup $K$ of $G$ such that $K'=G'\cap K$ is a maximal compact subgroup of $G'$. For a finite-length $\mathfrak{g}$-module $X$, write $\mathcal{V}_{\mathfrak{g}_{\mathbb C}}(X)\subseteq\mathfrak{g}_{\mathbb C}^*$ for its associated variety, and similarly write $\mathcal{V}_{\mathfrak{g}'_{\mathbb C}}(Y)\subseteq(\mathfrak{g}'_{\mathbb C})^*$ for a finite-length $\mathfrak{g}'$-module $Y$. Let $\operatorname{pr}_{\mathfrak{g}\to\mathfrak{g}'}:\mathfrak{g}_{\mathbb C}^*\to(\mathfrak{g}'_{\mathbb C})^*$ be the restriction map dual to the embedding $\mathfrak{g}'_{\mathbb C}\to\mathfrak{g}_{\mathbb C}$. Kobayashi's conjecture. If $X$ is an irreducible unitarizable $(\mathfrak{g},K)$-module and $Y$ is an irreducible $(\mathfrak{g}',K')$-module with $\operatorname{Hom}_{\mathfrak{g}'}(Y,X)\neq0$, then

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

The conjecture predicts that the associated variety of every irreducible constituent occurring in the restriction of an irreducible unitarizable representation is obtained by projecting the associated variety of the original representation. The paper confirms it for all discretely decomposable restrictions of minimal holomorphic representations to symmetric subgroups, while the general statement remains open.