Merkulov–Willwacher's injectivity conjecture for oriented graph cohomology

Let $d$ be an integer, and let $\textsf{OGC}_{2d+1}^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}$ be the oriented graph complex and $\textsf{RGC}_d^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}(\delta+\Delta_1)$ the ribbon graph deformation complex, with the morphism

$$\textsf{OGC}_{2d+1}^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}\longrightarrow \textsf{RGC}_d^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}(\delta+\Delta_1)$$

induced by the Chas–Sullivan morphism. Merkulov–Willwacher's conjecture. The induced map on cohomology is injective:

$$H^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}(\textsf{OGC}_{2d+1})\longrightarrow H^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}+1}(\textsf{RGC}_d(\delta+\Delta_1))$$

is injective. This concerns the relationship between oriented graph cohomology and the cohomology of ribbon graph complexes; the paper later indicates that the conjecture is proved using results relating these complexes to moduli spaces of curves.