The endo-parameter compatibility conjecture with the local Langlands correspondence

Let $G$ be the group under consideration, let $\operatorname{Irr}(G)$ denote its irreducible representations, let $LP_G(W_F)$ denote the Langlands parameters for $G$, and let $\mathcal{E}_{par}(F)$ denote the set of endo-parameters. Consider the diagram whose top horizontal map is the local Langlands correspondence, whose bottom horizontal map is the Bushnell–Henniart map, and whose vertical maps send a representation or parameter to its associated endo-parameter or restriction to the wild inertia group $P_F$.

Endo-parameter compatibility conjecture. The diagram relating $\operatorname{Irr}(G)$, $LP_G(W_F)$, $\mathcal{E}_{par}(F)$, and the representations of $P_F$ is commutative.

This conjecture predicts that endo-parameters extracted from full semisimple characters agree with the restrictions to wild inertia of the corresponding Langlands parameters, thereby relating the semisimple-character theory for inner forms of general linear groups to the local Langlands correspondence. Its status is not resolved in the supplied source.