Serre-functor conjecture for long twisting and shuffling functors

Let $\lambda$ be an integral parameter and let $\zeta$ be a generic cocharacter. Define the long twist and long shuffle by

$$\Phi^{\lambda,-\lambda}\circ\Phi^{-\lambda,\lambda},\qquad \Psi^{\zeta,-\zeta}\circ\Psi^{-\zeta,\zeta}.$$

These are endofunctors of the relevant bounded derived category. Serre-functor conjecture. The long twist and the long shuffle are both isomorphic to the right Serre functor, up to shift. This conjecture was subsequently proved by Losev.