Lewis–Reiner's Hurwitz orbit conjecture for reflection factorizations

Let $W$ be a well-generated finite complex reflection group, and let $T$ and $T'$ be reflection factorizations of a Coxeter element of $W$. Two factorizations are said to have the same multiset of conjugacy classes when the multisets formed by the conjugacy classes of their reflection factors coincide. Lewis–Reiner's conjecture. $T$ and $T'$ lie in the same Hurwitz orbit if and only if they share the same multiset of conjugacy classes. The conjecture predicts that these conjugacy-class data completely determine Hurwitz orbits of reflection factorizations of Coxeter elements; the paper confirms it for the complex reflection group $G_6$.