Kontsevich's formality theorem asserts that a certain quasi-isomorphism of chain complexes between the graded Lie algebra of polyvector fields on $\mathbb{R}^n$ and the dg Lie algebra of polydifferential operators (giving the Hoschchild-Kostant-Rozenberg theorem) can be extended to a $L_{\infty}$-quasi-isomorphism. This induces a bijection between equivalences classes of Poisson structures on $\mathbb{R}^n$ and equivalences classes of star products deforming the commutative product of $\mathcal{C}^{\infty}(\mathbb{R}^n)$ (the famous deformation quantization problem).
In his paper Formality conjecture, he introduced a certain graph complex (defined by linear combinations of isomorphisms classes of graphs). Obstructions to lifting the HKR quasi-isomorphism to a $L_{\infty}$-quasi-isomorphism live in the first cohomology group of this graph complex. The question to know wether this cohomology group is zero or not is still open. I would like to know what are the outcomes that a positive answer to this question would have (besides the annulation of theses obstructions).
