The consecutive-pair conjecture for order-three reflection groups

Let $W$ be a reflection group whose generators all have order three, let $c\in W$ be a Coxeter element, and let $T$ be a longer reflection factorization of $c$. A consecutive $(t,t)$ pair is a pair of adjacent equal reflections in a factorization. Consecutive-pair conjecture. The Hurwitz orbit of $T$ contains a reflection factorization with a consecutive $(t,t)$ pair. This is the proposed generalization of the corresponding lemmas for G4 and G5 and is intended to support the broader Hurwitz-transitivity conjecture; it is not proved here for the other order-three reflection groups discussed.