Endoscopic character identity for covers of reductive groups

Let $G$ be a reductive group with cover $G(F)_{x_G}$, let $(\dot s,\mathcal H)$ be a refined endoscopic datum for $G$, and let $H(F)_{x_H}$ be the corresponding endoscopic cover. Let $\varphi^G$ and $\varphi_H$ be tempered $L$-parameters related by the $L$-embedding $^L\xi_{H,G}$, and let $\Theta^{s,\mathsf{w}}_{\varphi_G}$ and $S\Theta_{\varphi_H}$ be the associated weighted and stable character distributions. If $f \in \mathcal C_c^\infty(G(F)_{x_G})$ and $f^H \in \mathcal C_c^\infty(H(F)_{x_H})$ are matching anti-genuine functions, then

$$\Theta^{s,\mathsf{w}}_{\varphi_G}(f)=S\Theta_{\varphi_H}(f^H).$$

Equivalently, for strongly regular semi-simple elements and the transfer factor $\Delta'_x[\mathsf{w}]$,

$$\Theta^{s,\mathsf{w}}_{\varphi_G}(\delta_G)=\sum_\gamma \Delta'_x[\mathsf{w}](\gamma_H,\delta_G)S\Theta_{\varphi_H}(\gamma_H),$$

where the sum runs over strongly regular semi-simple elements $\gamma$ of $H(F)$ up to stable conjugacy, and $\gamma_H\in H(F)_{x_H}$ is an arbitrary lift of $\gamma$. This is the expected character identity for refined endoscopic transfer in the setting of covers of reductive groups; the supplied text gives no evidence that it has been proved or disproved.