3-algebra conjecture for Ext of braided bialgebras

Let $A$ be a braided bialgebra. An algebra is a $3$-algebra when it is an algebra over the homology of the little $3$-cubes operad.

3-algebra conjecture for braided bialgebras. The graded algebra $\operatorname{Ext}_A^*(\mathbb{F},\mathbb{F})$ is a $3$-algebra.

This is presented as a conjecture related to Kontsevich's conjecture for Gerstenhaber-Schack cohomology. It extends the expected higher operadic structure from bialgebras to braided bialgebras.