Stable Bernstein center formulation of the endoscopic character identity

Let $(\mathbf{H},s,\eta)$ be an endoscopic triple for $\mathbf{G}$, and retain the cocharacter $\mu$, the half-sum of positive roots $\rho$, and the representation $r_{-\mu}$ introduced above. For $\tau\in \mathrm{Frob}^j I_E$ with $j\geq 1$ and $h\in C_c^\infty(\mathbf{G}(\mathbb{Z}_p))$, let $h^{\mathbf{H}}$ be an $\eta$-endoscopic transfer of $h$. Let $z_\tau^{\mathbf{H}}$ be the image in the Bernstein center of the stable Bernstein-center function sending a semisimple parameter $\lambda$ to

$$\operatorname{tr}\left(s^{-1}\tau\mid (r_{-\mu}\circ\eta\lambda|_{W_E})\right)|\cdot|_E^{-\langle\rho,\mu\rangle}.$$

Stable Bernstein center character identity conjecture. For any $\eta$-endoscopic transfer $h^{\mathbf{H}}$ of $h$, the function $z_\tau^{\mathbf{H}}\ast h^{\mathbf{H}}$ is a twisted endoscopic $\widetilde{\eta}$-transfer of $\phi_{\tau,h}$. This formulation is intended to extend the tempered character identity to nontempered parameters and expresses the transfer as factors depending separately on $\tau$ and $h$.