Relative Serre functor conjecture for Soergel bimodules of finite Coxeter systems

Let $(W,S)$ be a finite Coxeter system with longest element $w_0$, and choose a realization $\mathfrak{h}$ of $W$. Let $\mathbb{S}\mathrm{Bim}=\mathbb{S}\mathrm{Bim}(W,\mathfrak{h})$ be its category of Soergel bimodules. For $I\subset S$, let $\mathbb{S}\mathrm{Bim}_I$ be the full monoidal, idempotent-complete subcategory generated by the Bott–Samelson bimodules $B_s$ for $s\in I$. Write $w_I$ for the longest element of the parabolic subgroup $W^I$, let $\operatorname{FT}=F_{w_0}^{\otimes 2}$ and $\operatorname{FT}_I=F_{w_I}^{\otimes 2}$, and set

$$\operatorname{FT}_{S/I}=\operatorname{FT}\otimes \operatorname{FT}_I^{-1}=F_{v_I^{-1}}\otimes F_{v_I},$$

where $v_I$ is a shortest-length representative of the coset $w_0W_I$. Let $\pi_L,\pi_R:\mathbb{S}\mathrm{Bim}\to\mathbb{S}\mathrm{Bim}_I$ be the left and right adjoints to the fully faithful inclusion. Relative Serre functor conjecture. The complex $\operatorname{FT}_{S/I}$ tensor-commutes with all complexes in $\mathcal{K}^b(\mathbb{S}\mathrm{Bim}_I)$ up to natural homotopy equivalence, and

$$\pi_R(X)\simeq \pi_L(\operatorname{FT}_{S/I}\otimes X)\in\mathcal{K}^b(\mathbb{S}\mathrm{Bim}_I),$$

naturally in $X\in\mathcal{K}^b(\mathbb{S}\mathrm{Bim})$. The claim is presented as an expected generalization of the preceding type-$A$ result; its validity for arbitrary finite Coxeter systems is not established in the supplied text.