Cluster bounding-pair twists by Hurwitz moves

Determine whether there exist Hurwitz moves in the genus-three Smith pencil monodromy factorization that simultaneously cluster the bounding-pair twist products t_x t_z and t_b t_d.

Background

The genus–7 construction uses a positive factorization of a genus–3 Lefschetz pencil containing the bounding pairs (x,z) and (b,d). Because these pairs intersect intricately, the authors explain that ordinary Hurwitz moves do not immediately cluster both products in the desired way. They explicitly leave open whether some sequence of Hurwitz moves can achieve this simultaneous clustering, although their construction proceeds by exploiting a different configuration of the bounding pairs.