Presentation conjecture for friendly Calabi–Yau algebras of dimension three

Let $R$ be the base algebra, let $A$ be a friendly Calabi–Yau $R$-algebra of dimension $3$, let $F$ be a smooth $R$-algebra, and let $(\Omega^1_{\operatorname{nc},R}F)_{\operatorname{cyc}}$ denote the space of cyclic noncommutative $1$-forms. Write $\mathfrak A(F,\alpha)$ for the algebra associated with a cyclic $1$-form $\alpha$, and let conditions

be the conditions imposed in the source. **Presentation conjecture.** There \exists a smooth $R$-algebra $F$ and a cyclic $1$-form $\alpha\in(\Omega^1_{\operatorname{nc},R}F)_{\operatorname{cyc}}$ such that

A\cong\mathfrak A(F,\alpha),

and such that $\alpha$ satisfies conditions

. This is the more precise form of the claim that every Calabi–Yau algebra of dimension three has such a presentation. Its resolution status is not given in the source material.