Uniqueness of the isomorphism class of smooth sections in certain constructed examples

Ascertain whether the genus‑g Lefschetz fibrations over S^2 described in Example \ref{ex:surj-monodromy} have any smooth sections beyond those produced by the authors’ construction; equivalently, determine whether every smooth section is isomorphic to the constructed family so that these fibrations have a unique isomorphism class of smooth sections.

Background

The authors exhibit examples where their constructed sections (obtained via twisting along loops and fiber sums) all lie in a single isomorphism class. This suggests possible rigidity of section isomorphism classes in specific settings, notably when the monodromy representation is surjective onto Mod(Σ_{g,1}).

However, the authors cannot exclude the existence of additional sections not arising from their construction, which prevents concluding uniqueness of section isomorphism classes for those examples. Resolving this would clarify whether the observed single-orbit phenomenon genuinely reflects uniqueness or is an artifact of the construction.