Functorial braid group action on derived categories of moduli spaces

Let $X$ be the Salvetti complex associated with the discriminantal arrangement, let $\Theta$ be its set of vertices, and let $\Pi_1(X,\Theta)$ be the fundamental groupoid. For each vertex $\eta\in\Theta$, let $D^b(\mathfrak{M}_\eta)$ be the bounded derived category of the corresponding moduli space. For each oriented edge $\alpha:\eta\to\eta'$, let $\Phi_\alpha:D^b(\mathfrak{M}_\eta)\to D^b(\mathfrak{M}_\eta')$ be the associated Fourier--Mukai equivalence. Functorial braid group action conjecture. There exists a unique functor

$$\mathcal F:\Pi_1(X,\Theta)\longrightarrow \operatorname{Cat}$$

sending each vertex $\eta\in\Theta$ to the object $D^b(\mathfrak{M}_\eta)$ and each oriented edge $\alpha:\eta\to\eta'$ to the equivalence $\Phi_\alpha$. This would package the Fourier--Mukai equivalences associated with the oriented edges into a coherent action of the fundamental groupoid, with the codimension--2 relations ensuring compatibility of compositions along homotopic paths.