Prasad's conjecture on non-vanishing of twisted Jacquet modules

Let $F$ be the underlying local field, let $G_m=GL_m(F)$, and let $N$ and $\psi$ be the subgroup and character defining the twisted Jacquet module. Let $\pi$ be an irreducible smooth representation of $G_{2n}$ and let $\rho=\operatorname{rec}_F^{2n}(\pi)$ denote its Langlands parameter. Prasad's conjecture. The twisted Jacquet module $\pi_{N,\psi}$ vanishes if and only if the $L$-function

$$L(s,\pi,\operatorname{Ad})=L(s,\rho\otimes\rho^{\vee})$$

has poles of order at least $n,n-1,\dots,1$ at $s=1,2,\dots,n$, respectively. This conjecture classifies the irreducible smooth representations of $GL_{2n}(F)$ whose twisted Jacquet module is nonzero; its resolution is not indicated in the supplied text.