Equivalence of the Hurwitz transitivity and consecutive-pair conjectures

Let $W$ be a reflection group whose generators all have order three. Let $c\in W$ be a Coxeter element. Let $T$ be a longer reflection factorization of $c$. The Hurwitz transitivity conjecture asserts that reflection factorizations of $c$ are classified up to Hurwitz orbit by their multisets of conjugacy classes, while the consecutive-pair conjecture asserts that the Hurwitz orbit of $T$ contains a factorization with a consecutive $(t,t)$ pair. Equivalence conjecture. The Hurwitz transitivity conjecture and the consecutive-pair conjecture are logically equivalent. The paper presents this as a reduction for reflection groups with order-three generators; neither the general equivalence's constituent claims nor the resulting general transitivity statement is established there.