Endoscopic averaging formula for local Shimura cohomology

Let $G$ be a reductive group, let $\mu$ be a cocharacter with reflex field $E_{\mu}$, and let $(H,\mathcal{H},s,\eta)$ be an endoscopic datum such that an $L$-parameter $\phi$ factors as $\mathcal{L}_{F}\xrightarrow{\phi^{H}}\mathcal{H}\to{}^{L}G$. Let $R\Gamma^{\flat}_{c}(G,b,\mu)$ be the relevant cohomology complex, let $\mathrm{Red}^{\mathfrak e}_{b}$ be the reduction map, and let $S\Theta_{\phi^{H},H}$ be the associated stable distribution. Averaging Formula. One expects the equality in $K_{0}(G(\mathbb{Q}_{p})\times W_{E_{\mu}})$

$$\sum_{b\in B(G,\mu)}[R\Gamma^{\flat}_{c}(G,b,\mu)[\mathrm{Red}^{\mathfrak e}_{b}(S\Theta_{\phi^{H},H})]]=\mathrm{Trans}^{G}_{H}(S\Theta_{\phi^{H},H})\boxtimes\operatorname{tr}(r_{\mu}\circ\phi|_{W_{E_{\mu}}}\mid s).$$

For the trivial endoscopic datum $\mathfrak e_{\mathrm{triv}}=(G^{*},{}^{L}G^{*},1,\operatorname{id})$, this specializes to

$$\sum_{b\in B(G,\mu)}[R\Gamma^{\flat}_{c}(G,b,\mu)[\mathrm{Red}^{\mathfrak e_{\mathrm{triv}}}_{b}(S\Theta_{\phi,G^{*}})]]=S\Theta_{\phi,G}\boxtimes r_{\mu}\circ\phi|_{W_{E_{\mu}}}.$$

The paper describes this averaging formula as still largely conjectural and notes that the relevant stable-distribution hypothesis holds for all tempered $L$-parameters.