Trivalent partner conjecture for loop graphs in odd graph cohomology

Let $\mathrm{GC}_3$ be the graph complex, and let $L_k$ denote its loop graphs. The partner classes are those classes killed by the loop graphs in the spectral sequence described in the source. Trivalent partner conjecture. The classes killed by the loop graphs $L_k$ are linear combinations of trivalent graphs; equivalently, the corresponding classes live in degree $-3$ in $\mathrm{GC}_3$. The conjecture concerns the currently unknown partners of the loop graphs; the source reports that computationally found non-loop classes occur in degrees $-3$ and $-6$, and gives no resolution.