Gerstenhaber algebra conjecture for Ext groups of commuting central monoid-comonoid pairs

Let ${\mathscr{C}}$ be a monoidal category fulfilling a few mild conditions. Let $Z$ be a braided commutative monoid in the left weak monoidal centre of ${\mathscr{C}}$, and let $X$ be a braided cocommutative comonoid in the right weak monoidal centre of ${\mathscr{C}}$. A pair $(X,Z)$ is assumed to be a commuting pair. Gerstenhaber algebra conjecture. If $(X,Z)$ is a commuting pair, then

$$\operatorname{Ext}_{\mathscr{C}}(X,Z)$$

is a Gerstenhaber algebra. This conjecture proposes an enhancement of the known result that, under suitable exactness conditions, $\operatorname{Ext}_{\mathscr{C}}(\mathbb{1},\mathbb{1})$ carries a Gerstenhaber algebra structure; the stated enhancement is described as less known and so far unproven.