Nontriviality conjecture for the classes $X_{4k+2}$

For every $k=1,2,\dots$, let

$$X_{4k+2}=\pi\bigl([\hat L_{4k+1},\sigma_{4k+1}]\bigr)_{4k+2},$$

where $X_{4k+2}$ is the trivalent leading term described in the source, and let $\mathsf{GC}_2^{4k+2\text{-loop}}$ denote the corresponding loop-order component of the graph complex.

Nontriviality conjecture. For every $k=1,2,\dots$, the element $X_{4k+2}$ defines a non-trivial cohomology class in

$$H^{4k-1}\bigl(\mathsf{GC}_2^{4k+2\text{-loop}}\bigr).$$

The conjecture has been checked for $k=1$ and $k=2$. In particular, the class $X_{10}$ generates $H^7(\mathsf{GC}_2^{10\text{-loop}})$ in the checked case and survives to the stated spectral-sequence page; the general assertion remains open.