Generalized linear-period classification conjecture and Lapid–Prasad conjecture

Let $\mathrm{D}$ be a quaternion division algebra over a non-Archimedean local field $\mathrm{F}$ of characteristic zero, let $\mathrm{G}_n=\mathrm{GL}_n(\mathrm{D})$, and let $\mathrm{H}_{1,n-1}\simeq \mathrm{G}_1\times \mathrm{G}_{n-1}$ be the block-diagonal subgroup of $\mathrm{G}_n$. For $s\in\mathbb{R}$, define $\chi_s(\operatorname{diag}(g_1,g_2))=\nu(g_1)^{2s}\nu(g_2)^{-2s}$. The conjecture is that, for every $n>2$, an irreducible smooth representation $\pi$ of $\mathrm{G}_n$ satisfies $\operatorname{Hom}_{\mathrm{H}_{1,n-1}}(\pi,\chi_s)\neq 0$ if and only if its Langlands parameter $\mathfrak{L}(\pi)$ contains a Weil--Deligne subrepresentation isomorphic to $\mathfrak{L}(\nu^{-2s})$, of dimension $2n-4$, such that the resulting four-dimensional quotient is the Langlands parameter of either the trivial representation of $\mathrm{G}_2$, in which case $s=\pm\frac{n-2}{2}$, or an irreducible infinite-dimensional representation of $\mathrm{G}_2$ admitting an $\mathrm{H}_{1,1}$-invariant functional.