Rouquier's faithfulness conjecture for the 2-braid-group categorification

Let $B_{(W,S)}$ be the generalized braid group associated with the Coxeter system $(W,S)$, and let $2$-${\mathcal B}r$ be the monoidal category generated by the Rouquier complexes $E_s$ and $F_s$ for $s\in S$. Its Picard group, denoted $\operatorname{Pic}(2$-${\mathcal B}r)$, is the group of isomorphism classes of invertible objects under tensor product. Rouquier's faithfulness conjecture.

\operatorname{Pic}(2$-${\mathcal B}r)\cong B_{(W,S)}.

The source attributes this conjecture to Rouquier and notes that the underlying faithfulness statement was later proved for type $ADE$ by Brav–Thomas and for all finite types by Jensen; the displayed general identification is therefore presented in a context with substantial known resolution, but its exact status as formulated here is not established by the supplied evidence.