Asphericity of the fifth hyperpolygonal arrangement

Determine whether the complexified fifth hyperpolygonal arrangement is a K(π,1)-arrangement, thereby resolving the sole possible exception to the equivalence between freeness and the K(π,1) property within the family of hyperpolygonal arrangements.

Background

The paper proves that the hyperpolygonal arrangement $\mathcal H_n$ is free exactly for $n\leq 5$, and that it is not a K(π,1)-arrangement for $n\geq 6$. The arrangements for $n\leq 4$ are K(π,1) because they are simplicial, while the fifth arrangement $\mathcal H_5$ is free but falls outside the established simplicial range.

Consequently, the K(π,1) status of $\mathcal H_5$ is the only unresolved case needed to determine whether freeness and asphericity coincide throughout this family. Resolving it would also settle whether $\mathcal H_5$ satisfies the stated instance of Saito’s conjecture that freeness implies the K(π,1) property.