Trivial Lie bracket conjecture for Ext of braided bialgebras

Let $A$ be a braided bialgebra. The Gerstenhaber algebra structure on $\operatorname{Ext}_A^*(\mathbb{F},\mathbb{F})$ given by the cited theorem has trivial Lie algebra, meaning that its Lie bracket vanishes.

Trivial Lie bracket conjecture. The Lie algebra of this Gerstenhaber algebra is trivial.

The conjecture extends the vanishing result known in the paper for the Drinfeld double of a finite-dimensional Hopf algebra and, more generally, for cocommutative Hopf algebras.