Aspherical 4-manifolds admitting Lefschetz fibrations over S2

Determine whether a closed aspherical 4-manifold can admit a Lefschetz fibration over S^2.

Background

The paper constructs genus–9 Lefschetz fibrations whose total spaces admit compatible symplectically aspherical forms, but this does not establish that the total spaces themselves are aspherical. This motivates the broader question of whether asphericity of a closed 4-manifold is compatible with carrying a Lefschetz fibration over the 2-sphere.