Functorial invariance of parabolic-category tangle presentations

Let $D^b(\mathcal O^n)$ be the bounded derived category of the direct sum of the parabolic categories $\mathcal O^{k,n-k}$, and extend the cap, cup, and reflection functors to derived functors on these categories. For a presentation $\alpha$ of a tangle as a composition of elementary tangles, let $g(\alpha)$ be the corresponding composition of $\cap_{i,n}$, $\cup_{i,n}$, and $R_{i,n}$. Parabolic tangle-invariance conjecture. If $\alpha$ and $\beta$ are presentations of the same tangle $t$, then $g(\alpha)$ and $g(\beta)$ should be isomorphic, up to shifts in the derived category. This is the parabolic analogue of the preceding tangle-functor conjecture; no resolution is stated in the supplied text.