Conjecture on b[1mb[0mGL$1$\times GL$n-1$ distinction over a quaternion division algebra

Let $\operatorname{G}_n$ denote the relevant general linear group over a quaternion division algebra, and let $\operatorname{H}_{p,q}$ be the subgroup used to define linear periods, with $p+q=n$. A representation is $\operatorname{H}_{p,q}$-distinguished if it admits a nonzero $\operatorname{H}_{p,q}$-invariant linear form. For representations $\pi$ and $\tau$, write $\pi\times\tau$ for their parabolic induction, and let $\mathds{1}_m$ denote the trivial representation of $\operatorname{G}_m$.

Main conjecture. An irreducible smooth representation $\theta$ of $\operatorname{G}_n$ with $n>2$ is $\operatorname{H}_{1,n-1}$-distinguished if and only if $\theta$ is either $\mathds{1}_n$ or of the form

$$\mathds{1}_{n-2}\times\tau,$$

where $\tau$ is an infinite-dimensional irreducible representation of $\operatorname{G}_2$ which is $\operatorname{H}_{1,1}$-distinguished.

This conjecture aims to classify all irreducible smooth representations admitting a linear period with respect to $\operatorname{H}_{1,n-1}$ for $n>2$, extending the known classification in the case $n=2$. Earlier results establish multiplicity-one statements and exclude discrete series in the relevant setting, but the proposed classification remains unproved.