Braided monoidal structure conjecture for type A Soergel bimodule complexes

Let $m$ and $n$ denote objects indexing Rouquier complexes, and let $m\boxtimes n\rightarrow n\boxtimes m$ be the corresponding Rouquier complexes. Let $\tau_{m,n}$ be the naturality data given by bimodule morphisms. Write $\operatorname{K^b_{loc}}(\mathcal{D})$ for the locally graded $\mathbbm{k}$-linear semistrict monoidal $2$-category of complexes of diagrammatic Bott–Samelson bimodules of type A and chain maps up to homotopy. Braided monoidal structure conjecture. The Rouquier complexes $m\boxtimes n\rightarrow n\boxtimes m$, together with the naturality data provided by the bimodule morphisms $\tau_{m,n}$, equip $\operatorname{K^b_{loc}}(\mathcal{D})$ with a braided monoidal structure in the sense of the cited definition and Crans's slightly stricter notion. The conjecture asserts that the constructed braiding equivalences and naturality data satisfy the required coherence conditions for this braided monoidal $2$-category structure.