Arithmetic fundamental lemma for the spherical Hecke algebra

Let $\varphi'\in\mathcal{H}_{K'^{{{{{{{{\flat}}}}}}}}}\otimes\mathcal{H}_{K'}$, and let $\varphi=\operatorname{BC}(\varphi')\in\mathcal{H}_{K^{{{{{{{{\flat}}}}}}}}}\otimes\mathcal{H}_K$. Let $G'(F_0)_{\mathrm{rs}}$ and $G_{W_1}(F_0)_{\mathrm{rs}}$ denote the regular semisimple elements in the relevant groups, and let $\operatorname{Int}(g,\varphi)$ and $\operatorname{\partial Orb}(\gamma,\varphi')$ be the intersection number and derivative orbital integral. Arithmetic fundamental lemma. Whenever $\gamma\in G'(F_0)_{\mathrm{rs}}$ is matched with $g\in G_{W_1}(F_0)_{\mathrm{rs}}$,

$$2\operatorname{Int}(g,\varphi)\cdot\log q=-\omega(\gamma)\operatorname{\partial Orb}(\gamma,\varphi').$$

This is the central AFL identity for the spherical Hecke algebra; it is proved in the paper when $n=1$, while the general case remains open.