Transitivity conjecture for the braid-group action on $\mathrm{X}^{(2)}_{\gamma,\delta}$

Let $B_4$ be the braid group on four strands, and let $\mathrm{X}^{(2)}_{\gamma,\delta}$ be the set on which the induced braid-group action is defined. Assume Assumption. Transitivity conjecture. Under this assumption, the action of $B_4$ on $\mathrm{X}^{(2)}_{\gamma,\delta}$ is transitive. This conjecture is proposed as an ingredient for proving that the alternating quotients have girth tending to infinity; the paper also notes that a weaker statement, that a subset of density tending to $1$ lies in one orbit, may suffice.