Existence of 2-sections for all Lefschetz fibrations

Establish whether every Lefschetz fibration over S^2 admits a 2-section.

Background

The paper proves that every Lefschetz fibration over S^2 admits positive multisections of sufficiently divisible degree, with the allowable degrees determined by the divisibility of the fiber class. The authors then ask whether degree 2 always suffices. By their multisection-degree theorem, this is equivalent to asking whether the fiber-class divisibility always divides 2; primitivity of every fiber class would imply the stronger conclusion that every degree at least 2 occurs.