The Epsilon Dichotomy Conjecture for strongly tempered spherical models

Fix a local field $F$, a model $(G,H)$ in the paper's table, a tempered $L$-parameter $\phi$, its component group $S_\phi$, and the representation $\rho_X$ attached to the spherical variety. For an elliptic extended endoscopic triple lifting $s\in S_\phi$ to $s'$, let $\omega_{\phi,H}(s')$ be the relevant local epsilon factor, with the additional sign in the two unitary cases as specified in the paper. Epsilon Dichotomy Conjecture. The function $\omega_{\phi,H}$ is independent of the choice of elliptic extended endoscopic triple and is a quadratic character of $S_\phi$; moreover, the unique $(H,\chi)$-distinguished member of the $L$-packet $\Pi_\phi$ is the member associated with $\omega_{\phi,H}$. The conjecture predicts both well-definedness of the epsilon-character and the precise distinguished representation; no resolution status is supplied in the source.