to an isogeny class of abelian varieties with $G$-structure, and the set $S(\phi)$ is to parameterize the isomorphism classes in the corresponding isogeny class. More precisely, $X_p(\phi)$ and $X^p(\phi)$ should correspond, repsectively, to the isogenies of $p$-power order and prime-to-$p$ order (say, leaving from a fixed member in the isogeny class $\phi$) preserving $G$-structure, and $X_p(\phi)$ can be also identified with a suitable affine Deligne-Lusztig variety $X(\{\mu_X\},b)_{{\mathbf{K}}_p}$. The term $I_{\phi}({\mathbb Q})$ is to be the automorphism group of the isogeny class attached to $\phi$, and thus acts naturally on $X_p(\phi)$ and $X^p(\phi)$. Moreover, each of the sets $S(\phi)$ carries a compatible action of the group $G({\mathbb A}_f^p)$ and the Frobenius automorphism $\Phi$ (which is an element of ${\mathrm{Gal}}({\bar{\mathbb{F}}_{p}}/\kappa(\wp))$), and the bijection (\[eqn:LRconj-ver1\]) should be compatible with these actions.

When it comes to the precise definition, the most tricky object is the parameter $\phi$. Its precise definition makes use of the language of *Galois gerb*: A Galois gerb is a gerb, in the sense of *Cohomologie non abélienne* à la Giraud, on the étale site of a field (with choice of a neutralizing object). This is motivated by the fact [@Milne94] that there is a well-determined class of Shimura varieties (i.e. 