Hurwitz transitivity conjecture for Singer-cycle factorizations

Let $c$ be a Singer cycle in $GL_n({\mathbb{F}}_q)$, and consider ordered factorizations $c=t_1t_2\cdots t_\ell$ into reflections. The multiset $\{\det(t_i)\}_{i=1}^\ell$ is an invariant of the Hurwitz action of the braid group on these factorizations. Hurwitz transitivity conjecture. Any two factorizations $c=t_1 t_2 \cdots t_\ell$ with the same multiset $\{\det(t_i)\}_{i=1}^\ell$ lie in the same Hurwitz orbit. In particular, there is only one Hurwitz orbit of factorizations when $q=2$ for any $\ell$. This is an analogue of the transitivity theorem for shortest factorizations of Coxeter elements in well-generated complex reflection groups; the determinant multiset is the evident obstruction when $q\ne 2$.