Relationship between the cohomologies of GC₂ and GC₃

Determine whether there exists a substantial structural link between the cohomology of the even Kontsevich graph complex GC₂ and the cohomology of the odd Kontsevich graph complex GC₃, beyond the observed similarity of Euler characteristics.

Background

The complexes GC_n for even and odd n are closely related via loop-order-dependent regrading, and their Euler characteristics exhibit similarities.

Despite these parallels, the authors explicitly note the absence of any known substantial link between the cohomologies of GC₂ and GC₃, highlighting a foundational structural question.