I consider the fundamental lemma for the spherical Hecke algebra.

Let $G$ a connected reductive quasisplit group on $F$, a local field of equal characteristic $p$. and $H$ an endoscopic group.

Can we reduce the fundamental lemma for (G,H) to the fundamental lemma for $(H,G)$ with $G_{der}=G_{sc}$ or even better to (H, G) where both G and H have a simply connected derived group?