Asphericity of the K4 connected subgraph arrangement

Determine whether the connected subgraph arrangement associated with the complete graph K4 is K(π, 1).

Background

The paper proves that every aspherical connected subgraph arrangement is free, with the sole possible exception of the arrangement associated with K4. Because the K4 arrangement is not free, the theorem does not resolve its topological status; its K(π, 1) property remains unknown.

References

While AK4 is not free, it is unknown at present whether AK4 is K(π, 1).

— On connected subgraph arrangements  (2502.18144 - Giordani et al., 25 Feb 2025) in Immediately after Theorem 1.8, Section 1, p. 4