Prasad–Takloo-Bighash conjecture for distinction of discrete series

Let $F$ be a non-archimedean local field of characteristic not $2$, let $E/F$ be a quadratic extension, let $D$ be a central division $F$-algebra of dimension $d^2$, and let $A=\mathcal{M}_n(D)$. Write $C_{\mathcal{M}_n(D)}(E)$ for the centralizer of $E$ in $\mathcal{M}_n(D)$, and let $N_{rd,F}$ and $N_{rd,E}$ denote the relevant reduced norms. Let $\pi$ be an irreducible admissible representation of $A^\times$ whose Jacquet–Langlands transfer to $GL(nd,F)$ is generic, with central character $\omega_\pi$. Let $\mu$ be a character of $E^\times$ satisfying

$$\mu^{\frac{nd}{2}}|_{F^\times}=\omega_\pi.$$

Assume that $\mu\circ N_{rd,E}$ appears as a quotient of the restriction of $\pi$ to $C_{\mathcal{M}_n(D)}(E)^\times$. Prasad–Takloo-Bighash conjecture. Then the Langlands parameter of $\pi$ takes values in $GSp_{nd}(\mathbb{C})$ with similitude factor $\mu|_{F^\times}$, and

$$\epsilon\left(\frac{1}{2},\pi\otimes\operatorname{Ind}_E^F(\mu^{-1})\right)=(-1)^n\omega_{E/F}(-1)^{\frac{nd}{2}}\mu(-1)^{\frac{nd}{2}},$$

where $\omega_{E/F}$ is the quadratic character of $F^\times$ whose kernel is the norm subgroup of $E^\times$. If $\pi$ is a discrete-series representation of $A^\times$, these two conditions are necessary and sufficient for $\mu\circ N_{rd,E}$ to appear as a quotient of the restriction of $\pi$ to $C_{\mathcal{M}_n(D)}(E)^\times$. The paper proves this conjecture for Steinberg representations, while the general assertion for discrete series is the remaining conjectural statement.