Kontsevich's 3-algebra conjecture for Gerstenhaber-Schack cohomology

Let $H$ be a bialgebra. Denote by $H_{GS}(H,H)$ its Gerstenhaber-Schack cohomology, and let $\mathcal{C}_3$ be the little $3$-cubes operad.

Kontsevich's conjecture. The graded algebra $H_{GS}(H,H)$ is a $3$-algebra, that is, an algebra over the homology of the little $3$-cubes operad $\mathcal{C}_3$.

The conjecture is attributed to Kontsevich and is stated in the paper as known to have an announced proof when $H$ is a Hopf algebra; the bialgebra case remains open in the source.