Prasad–Takloo-Bigheash local distinction conjecture

Let $F_v$ be a local field, let $D_v$ be a quaternion division algebra over $F_v$, and let $H_v$ and $H'_v$ be the relevant subgroups of $\operatorname{GL}_n(D_v)$ and $\operatorname{GL}_{2n}(F_v)$. Let $\pi_v$ and $\pi'_v$ be irreducible admissible representations of these two groups, respectively, and let $\pi_{E,v}$ and $\pi'_{E,v}$ denote their respective base changes. Prasad–Takloo-Bigheash conjecture. If $\pi_v$ (respectively, $\pi'_v$) is $H_v$-distinguished (respectively, $H'_v$-distinguished), then it is symplectic and satisfies

$$\epsilon(1/2,\pi_{E,v})=(-1)^n \quad\text{(respectively, }\epsilon(1/2,\pi'_{E,v})=1\text{)}.$$

Conversely, these conditions are sufficient for distinction when $\pi_v$ (respectively, $\pi'_v$) is a discrete-series representation. This local conjecture predicts the epsilon-factor criterion underlying local distinction and dichotomy; the source notes that a more general statement was conjectured and that the case $n=2$ had been established using the local theta correspondence.