The generalized epsilon dichotomy conjecture for the GL6 model

Let $G=\operatorname{GL}_6$, $H=\operatorname{GL}_2\ltimes U$, and let $\chi'$ be defined by $\chi'(\operatorname{diag}(h,h,h)u)=\alpha(\det h)\xi(u)$. Let $\pi$ be irreducible tempered with central character $\alpha^2$, let $\phi_\pi$ be its Langlands parameter, let $\phi_\alpha$ correspond to $\alpha$, and let $\pi_D$ be its Jacquet–Langlands transfer when it exists, otherwise $0$. The generalized epsilon dichotomy conjecture.

$$m(\pi,\chi')=1\iff\epsilon\left(1/2,(\rho_X\circ\phi_\pi)\otimes\phi_\alpha^{-1}\right)=1,$$ $$m(\pi_D,\chi_D')=1\iff\epsilon\left(1/2,(\rho_X\circ\phi_\pi)\otimes\phi_\alpha^{-1}\right)=-1.$$

This generalizes the GL6 epsilon dichotomy to the character $\chi'$; the source supplies no resolution evidence.