The character identity conjecture for disconnected groups

Let $\phi:L_F\to{}^LG$ be a tempered Langlands parameter, let $\tilde s\in\tilde S_\phi^{[z]}$ be semisimple, and let $\mathfrak e$ be the associated endoscopic datum. Choose a $z$-pair and write $\phi^{\mathfrak z}$ for the induced parameter. Let $\tilde\Pi_{\phi,z}$ be parametrized by $\operatorname{Irr}(\pi_0(\tilde S_\phi^{[z]}),[z])$, with $\tilde\rho\mapsto\tilde\pi_{\tilde\rho}$. The character identity conjecture. For matching functions $f$ and $f^{\mathfrak z}$, one has

$$S\Theta_{\phi^{\mathfrak z}}(f^{\mathfrak z})=\sum_{\tilde\rho}\operatorname{tr}(\tilde\rho)(\tilde s)\,\Theta_{\tilde\pi_{\tilde\rho}}(f),$$

where the sum runs over $\operatorname{Irr}(\pi_0(\tilde S_\phi^{[z]}),[z])$. This is the expected endoscopic character relation accompanying the local Langlands parametrization; its validity depends on the conjectural packets and transfer factors for disconnected groups.