Prasad–Takloo-Bighash reformulation via component-group characters

Assume $F$ is non-archimedean. Let $\phi\colon WD_F\to\mathrm{GL}_n(\mathbb{C})$ be an $L$-parameter, let $\Pi_\phi$ be the packet across all inner forms of $\mathrm{GL}_n(F)$, and let $S_\phi=Z_{\mathrm{SL}_n(\mathbb{C})}(\phi)/Z_{\mathrm{SL}_n(\mathbb{C})}(\phi)^\circ$. Write

$$\phi=\bigoplus_{i\in I_\phi}\phi_i\oplus\bigoplus_{j\in J_\phi}(\phi_j\oplus(\phi_j^\vee\cdot\chi|_{F^\times})),$$

where each $\phi_k\colon WD_F\to\mathrm{GL}_{n_k}(\mathbb{C})$ is discrete, each $n_i$ is even, each $\phi_i$ is symplectic similitude with factor $\chi|_{F^\times}$, and the $\phi_i$ are pairwise nonisomorphic. If $\Pi_\phi$ is generic, Prasad–Takloo-Bighash reformulation. For $\pi\in\Pi_\phi$, $m(\pi,\chi)\neq0$ if and only if

$$\varepsilon\left(\phi_i\otimes\operatorname{Ind}_{W_E}^{W_F}(\chi^{-1})\right)=\left(\chi_\pi(\zeta_n)\chi\eta_{E/F}(-1)\right)^{\frac{n_i}{2}},\qquad i\in I_\phi,$$

where $\zeta_n=\exp(2\pi\sqrt{-1}/n)$ is a primitive $n$-th root of unity. This reformulates distinction in terms of component-group characters and epsilon factors; the supplied text gives no resolution status.