The equivalence of infinitesimal and unitary discrete decomposability

Let $G$ be a real reductive Lie group, let $G'$ be a reductive subgroup, and let $\pi$ be an irreducible unitary representation of $G$. Write $\pi|_{G'}$ for its restriction, and call it infinitesimally discretely decomposable when the underlying $(\mathfrak{g},K)$-module $\pi_K$ is discretely decomposable as a $\mathfrak{g}'$-module.

Discrete decomposability equivalence conjecture. The following two conditions on $(G,G',\pi)$ are equivalent: (i) the restriction $\pi|_{G'}$ is infinitesimally discretely decomposable; (ii) the unitary representation $\pi$ decomposes discretely into a direct sum of irreducible unitary representations of $G'$.

The converse implication (ii) $\Rightarrow$ (i) remains open, while affirmative results have been partially obtained by Duflo and Vargas for discrete series representations. Thus the equivalence is not established in general.