Inductive construction conjecture for discrete series of metaplectic and quaternionic orthogonal groups

Let $A_{p,q}(k)$ denote a middle-dimensional cohomologically induced representation associated with a Levi subgroup $\widetilde{U}(p,q)$ of $Mp_{2n}(\mathbb R)$ or with a Levi subgroup $U(p,q)$ of $O^*(2n)$. Let $\mathcal H_\sigma$ be the Hilbert space of a discrete series representation $\sigma$ of the corresponding smaller group. Inductive construction conjecture. Every discrete series representation of $Mp_{2n}(\mathbb R)$ can be constructed inductively by an invariant tensor product

$$\mathcal H_\sigma\otimes_{Mp_{2n-2}(\mathbb R)}A_{p,q}(k)^\infty$$

for some $k,p,q$ with $p+q=2n-1$, where $\sigma$ is a discrete series representation of $Mp_{2n-2}(\mathbb R)$. Likewise, every discrete series representation of $O^*(2n)$ can be constructed inductively by

$$\mathcal H_\sigma\otimes_{O^*(2n-2)}A_{p,q}(k)^\infty$$

for some $k,p,q$ with $p+q=2n-1$, where $\sigma$ is a discrete series representation of $O^*(2n-2)$. This is proposed as an extension of the proved inductive construction for $U(p,q)$; the paper does not establish these assertions.