The $t$-exactness conjecture for the endoscopic equivalence

Let $\mathbf{G}$ be the reductive group, let $\mathrm{F}$ be its Frobenius morphism, and let $\ell$ be the coefficient characteristic. Let the endoscopic equivalence refer to the equivalence constructed in Theorem

of the source, between the categories appearing there. **$t$-exactness conjecture.** The endoscopic equivalence of Theorem

is $t$-exact.

The source proves $t$-exactness under a hypothesis on the affineness of the relevant Deligne–Lusztig varieties and suggests that this hypothesis is superfluous. In the unipotent case with $\overline{\mathbb{Q}}_{\ell}$ coefficients, the needed cohomological bound is cited as known, but the stated general conjecture remains open.