Nontriviality of the classes X_{4k+2} in GC₂

Prove that for every integer k ≥ 1, the element X_{4k+2} = π([\hat L_{4k+1}, σ_{4k+1}])_{4k+2}, where \hat L_{4k+1} is a (δ+∇)-primitive of σ_{2k+1} and π is the projection to trivalent graphs, defines a nontrivial cohomology class in H^{4k−1}(GC₂^{4k+2-loop}).

Background

In Section 5, the authors construct explicit cocycles X_{4k+2} by taking leading trivalent parts of brackets [\hat L_{4k+1}, σ{4k+1}], where \hat L{4k+1} is chosen so that (δ+∇)\hat L_{4k+1} equals σ_{2k+1} up to higher-loop terms.

They verify the nontriviality for k=1 by a spectral-sequence argument and by computer for k=2, and then formulate the general conjecture asserting nontriviality for all k.