Prasad–Takloo-Bighash epsilon dichotomy conjecture for linear periods

Let $F$ be a finite extension of $\d075\mathbb{Q}_p$ or $F=\mathbb{R}$, let $D$ be a central division algebra over $F$ with $\dim_F(D)=d^2$, put $G=\mathrm{GL}_m(D)$ and $n=md$, and let $E/F$ be quadratic. Assume that $n$ is even, let $H$ be the centralizer of $E^\times$ in $G$, and let $m(\pi,\chi)$ denote the multiplicity of the $(H,\chi_H)$-period for a representation $\pi$ of $G(F)$. For an $L$-parameter $\phi\colon WD_F\to\mathrm{GL}_n(\mathbb{C})$ and a character $\chi$ of $E^\times$, Prasad–Takloo-Bighash conjecture. If $m(\pi,\chi)\neq0$, then $\phi$ takes values in $\mathrm{GSp}_n(\mathbb{C})$ with similitude factor $\chi|_{F^\times}$ and

$$\varepsilon\left(\phi\otimes\operatorname{Ind}_{W_E}^{W_F}(\chi^{-1})\right)=(-1)^m\chi\eta_{E/F}(-1)^{\frac{n}{2}}.$$

For $\pi\in\operatorname{Irr}_\mathrm{disc}(G(F))$, the converse holds: these two conditions imply $m(\pi,\chi)\neq0$. Here $\chi|_{F^\times}$ is viewed as a character of $W_F$ by local reciprocity. This is the original epsilon-dichotomy formulation of the Prasad–Takloo-Bighash conjecture; its status is not specified in the supplied text.