Degree-two spectral-gap reduction for random walks on the unitary group

Prove the general degree-two spectral-gap reduction conjectured by Alon and Puder for random walks on the unitary group generated by weighted hypergraphs.

Background

The paper proves a degree-two spectral-gap reduction for a broad class of stochastic exchange dynamics, and notes that this settles a related conjecture of Alon and Puder. It explicitly distinguishes that result from the more general unitary-group conjecture, which is not addressed by the paper.

The unresolved problem concerns determining whether the spectral gap for the relevant weighted-hypergraph-generated random walks on the unitary group is always realized in degree at most two.