The Conformal Index conjecture for braid types of routes to chaos

Let $C$ and $C'$ be different routes to chaos, and let their Conformal Index be the endpoint invariant associated with each route. Say that $C$ and $C'$ define the same braid types when the collections of braid types arising from the two routes agree. Conformal Index conjecture. The routes $C$ and $C'$ have the same Conformal Index precisely when $C$ and $C'$ define the same braid types. The paper states that this would characterize the braid-type data detected by the Conformal Index, but gives no proof and explicitly says that the conjecture is not solved.