The obstruction to achiral Lefschetz fibrations on closed four-manifolds

A broken Lefschetz fibration (BLF) is a fibration allowing Lefschetz singularities and round $1$-handle singularities, while an achiral fibration also allows anti-Lefschetz singularities. A closed, smooth, oriented $4$-manifold is called BLF-admitting if it admits a BLF over $S^2$.

BLF achirality conjecture. Not all closed, smooth, oriented $4$-manifolds admit BLFs; in particular, it is possible that $\mathbb{C}P^2$ is necessarily achiral as a fibration, even though it has a Lefschetz pencil structure.

The source presents this as a possibility rather than a resolved assertion, so its status is unclear from the supplied text. The claim concerns the distinction between broken and achiral Lefschetz fibrations on four-manifolds.