The graph-homology identification conjecture for diagram cohomology

For each $m$, let ${\cal D}^*_m$ be the dual diagram complex with differential $\partial$, and let $H^1_\partial({\cal D}^*_m)$ denote its first cohomology group. The graph-homology identification conjecture. The group $H^1_\partial({\cal D}^*_m)$ is isomorphic to a certain twist of Kontsevich's graph homology. This is offered as an alternative diagram-level description related to the Fundamental Theorem; the source does not specify the twist further or give a proof.