Let $G = SO_5(\mathbf{Q}_p)$ be the split special orthogonal group over $\mathbf{Q}_p$, and $H \subset G$ the group $SO_4(\mathbf{Q}_p)$, embedded in the usual way as the stabiliser of an anisotropic vector.

Let $\pi$ be an irreducible unramified principal series representation of $G$. Is $Hom_H( \pi \mid_H, \mathbf{C})$ non-zero?

(NB: I'm aware that the Gan--Gross--Prasad conjectures give all sorts of insights into the spaces $Hom_H( \pi \mid_H, \tau)$ where $\tau$ is an irreducible rep of $H$. But they always seem to assume $\tau$ is generic, which the trivial representation is not.)

Let $G = SO_5(\mathbf{Q}_p)$ be the split special orthogonal group over $\mathbf{Q}_p$, and $H \subset G$ the group $SO_4(\mathbf{Q}_p)$, embedded in the usual way as the stabiliser of an anisotropic vector.

Let $\pi$ be an irreducible unramified principal series representation of $G$. Is $Hom_H( \pi \mid_H, \mathbf{C})$ non-zero? (That is, is $\pi$ necessarily $H$-distinguished?)

(NB: I'm aware that the Gan--Gross--Prasad conjectures give all sorts of insights into the spaces $Hom_H( \pi \mid_H, \tau)$ where $\tau$ is an irreducible rep of $H$. But they always seem to assume $\tau$ is generic, which the trivial representation is not.)