Asphericity of the original genus-9 total space

Determine whether the original genus–9 Lefschetz-fibration total space X=X_1 is aspherical.

Background

The genus–9 example X=X_1 admits compatible symplectically aspherical symplectic forms, meaning that the forms evaluate trivially on spherical classes. The authors distinguish this property from topological asphericity of the manifold and explicitly state that they do not know whether X itself is aspherical. They show that the untwisted self fiber sums X_n for n≥2 are not aspherical because their fundamental groups contain nontrivial finite cyclic subgroups.