The well-defined epsilon-character conjecture for strongly tempered spherical varieties

Let $\phi':W_F'\rightarrow {}^LG_X$ be a tempered Langlands parameter, and let $Z_{\phi'}$ be its centralizer with component group $S_{\phi'}=Z_{\phi'}/Z_{\phi'}^{\circ}$. For $s\in Z_{\phi'}$, choose an extended endoscopic triple $(G',s,{}^L\eta)$ through which $\phi'$ factors and a lifting $\phi_0$ with $\phi'={}^L\eta\circ\phi_0$. Let $\omega_{\phi',\rho_X}(s)=\eta\circ\phi_0(-1)\epsilon(\frac{1}{2},\rho_{X,s,{}^L\eta,-}\circ\phi_0)$.

The epsilon-character conjecture. The function $\omega_{\phi',\rho_X}$ is well defined, independent of the extended endoscopic triple and the lifting, descends to a function on $S_{\phi'}$ by being constant on connected components of $Z_{\phi'}$, and is a character of $S_{\phi'}$.

This conjecture supplies the quadratic character used in the paper's proposed description of distinguished representations in tempered $L$-packets. The paper assumes it after observing that independence of the auxiliary choices is not automatic.