The Hurwitz transitivity conjecture for reduced reflection factorizations

Let $B$ be the braid group associated with a well-generated complex reflection group, let $\mathcal{R}$ be its set of braid reflections, and let $\delta$ be the braid-group analogue of a Coxeter element. Write $\operatorname{Red}_{\mathcal{R}}(\delta)$ for the reduced factorizations of $\delta$ into elements of $\mathcal{R}$. Hurwitz transitivity conjecture. The Hurwitz action of $B_n$ on $\operatorname{Red}_{\mathcal{R}}(\delta)$ is transitive. The claim is intended as the braid-group analogue of the known transitivity of the Hurwitz action on reduced reflection factorizations in the reflection group; the paper gives no resolution here.