Relative Serre duality conjecture for parabolic induction

Let $(W,S)$ be a Coxeter system, let $I\subseteq S$ be a subset of simple reflections, and let $W_I$ be the parabolic subgroup generated by $I$. Let $\mathfrak{h}$ be a realization of $W$, viewed also as a realization of $W_I$, and let

$$\iota:D^m(\mathfrak{h},W_I)\to D^m(\mathfrak{h},W)$$

be the monoidal parabolic-induction functor, with left and right adjoints $\iota^L$ and $\iota^R$. Let $\operatorname{FT}_{W,I}\in D^m(\mathfrak{h},W)$ be the relative full twist defined using Rouquier's braid-group action. Relative Serre duality conjecture. If $W$ is finite, there are natural isomorphisms of functors

$$\iota^L(\operatorname{FT}_{W,I}\star -)\cong\iota^R\cong\iota^L(-\star\operatorname{FT}_{W,I}),$$

and $\operatorname{FT}_{W,I}$ naturally commutes under $\star$ with objects of $D^m(\mathfrak{h},W_I)$. The conjecture was proved in the cited source, so the relative full twist describes both adjoints to parabolic induction and is central relative to the parabolic mixed derived category.