Twisted Jacquet module conjecture for cuspidal representations of GL(2n)

Let $F_n$ be the unique degree-$n$ extension of $F$, let $G=\operatorname{GL}(2n,F)$, and let $A\in\operatorname{M}(n,F)$ be the matrix specified in the source. Let $H_A=M_1\times M_2$ and $U_A=U_1\times U_2$, where $M_1,M_2,U_1,U_2$ are the corresponding subgroups of $\operatorname{GL}(n,F)$, and let $N$, $\psi_A$, $M_{\psi_A}$, and $\mu$ be as defined in the source. For a regular character $\theta$ of $F_n^{\times}$, let $\pi=\pi_\theta$ be the associated irreducible cuspidal representation of $G$. Twisted Jacquet module conjecture. Then

$$\pi_{N,\psi_A}\simeq \theta|_{F^{\times}}\otimes\operatorname{ind}_{U_A}^{H_A}\mu$$

as $M_{\psi_A}$-modules. This conjecture is motivated by explicit calculations for $\operatorname{GL}(4,F)$ and $\operatorname{GL}(6,F)$; its status is not determined by the supplied source context.