Functorial transfer conjecture for cohomological representations from $\operatorname{Sp}(4,\mathbb{R})$

Let $\pi$ be an irreducible unitary representation of $\operatorname{Sp}(4,\mathbb{R})$ such that $\pi$ has non-vanishing cohomology, and let $\iota(\pi)$ denote its transferred representation to $\operatorname{GL}(5,\mathbb{R})$. Let $M_\lambda$ be a finite-dimensional representation such that $\pi$ is cohomological with respect to $M_\lambda$, and let $\iota(M_\lambda)$ denote the corresponding transferred coefficient representation.

Functorial transfer conjecture. The representation $\iota(\pi)$ is cohomological if $\pi$ is one of the following: the trivial representation; a discrete series representation of $\operatorname{Sp}(4,\mathbb{R})$; or a representation induced from the Siegel parabolic. Furthermore, $\iota(\pi)$ is cohomological with respect to $\iota(M_\lambda)$ whenever $\pi$ is cohomological with respect to $M_\lambda$.

The conjecture would complete the classification of which unitary irreducible cohomological representations of $\operatorname{Sp}(4,\mathbb{R})$ transfer to cohomological representations of $\operatorname{GL}(5,\mathbb{R})$, including non-trivial coefficient systems. The paper notes that the corresponding classification in the non-trivial coefficient case is difficult because an analogous version of Speh's classification is unavailable.