Chinello–Dat conjecture on endo-parameter blocks for inner forms

Let $({\rm G},h)$ be an inner form of a general linear group or of a classical group with $p eq 2$, and let $\frak{t}$ be an endo-parameter for ${\rm G}$. Let ${\rm R}$ be an integral domain and a $\bbZ[1/p,bc_{p^infty}]$-algebra. ChinelloDat conjecture. There is an equivalence of categories

$$\operatorname{Rep}_{{\rm R}}(\mathfrak{t})\simeq \operatorname{Rep}_{{\rm R}}(\mathbf{0}_{{\rm G}_{\beta}}).$$

This conjecture generalizes a result of Chinello and is related to predictions of Dat; it proposes that the category attached to an endo-parameter is equivalent to the category associated with the zero endo-parameter for the corresponding group.