The splendid Morita equivalence conjecture for 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

$$of the source. A **splendid Morita equivalence** is understood in the sense of Rickard. **Splendid Morita equivalence conjecture.** The endoscopic equivalence of Theorem$$

induces a splendid Morita equivalence in the sense of Rickard.

This conjecture asks for a stronger algebraic realization of the categorical endoscopic equivalence. The source presents it among possible conjectural strengthenings and gives no resolution, so it remains open.