Characterization of freeness by free cumulants in the odd-degree case

Prove that vanishing mixed free cumulants characterizes tensorial freeness for families of tensors when odd degrees appear in the associated non-crossing poset, or otherwise develop an appropriate cumulant framework that yields such a characterization in the odd case.

Background

The paper introduces a non-crossing poset on trace maps and defines free cumulants via Möbius inversion. For even degrees, the authors established that vanishing mixed cumulants characterize freeness, paralleling classical free probability.

When odd degrees appear, uniqueness of minimal elements in the poset fails, obstructing the standard argument. The authors explicitly note they could not prove the characterization in this odd case and suggest a possible remedy via multiplicities, but the result remains unproven.