Explicit adjoint formulas for parabolic Soergel bimodule categories

Let $(W,S)$ be a Coxeter system, let $I\subset S$, and let $\mathfrak{h}$ be a realization of $W$ with fundamental weights $\{\rho_s\}_{s\in S}$. Write $\{s_1,\ldots,s_\ell\}$ for the complement of $I$ in $S$. Let $\pi_L$ and $\pi_R$ be the left and right adjoints to the inclusion

$$K^b(\mathbb{S}\operatorname{Bim}(\mathfrak{h},W_I))\longrightarrow K^b(\mathbb{S}\operatorname{Bim}(\mathfrak{h},W)).$$

Explicit adjoint formulas conjecture. The adjoints satisfy

$$\pi_L\cong \pi^+_{s_1}\circ\cdots\circ\pi^+_{s_\ell},\qquad \pi_R\cong \pi^-_{s_1}\circ\cdots\circ\pi^-_{s_\ell}.$$

These formulas are independent of the order chosen for the complement of $I$ in $S$. The conjecture gives an explicit description of the adjoints whose existence is known in general, but the asserted formulas and order-independence remain open.