Formality conjecture for dg multiplicative preprojective algebras

Let $Q$ be a connected quiver, let $q\in(k^\times)^{Q_0}$, let $\Lambda^q(Q)$ be its multiplicative preprojective algebra, and let $\Lambda^{\mathrm{dg},q}(Q)$ be the dg multiplicative preprojective algebra. The paper proves formality in the case of quivers containing a cycle.

Formality conjecture. If $Q$ is connected and not Dynkin, then

$$\Lambda^{\mathrm{dg},q}(Q)\text{ is quasi-isomorphic to }\Lambda^q(Q),\text{ concentrated in degree zero}.$$

Formality identifies the dg algebra arising from the Fukaya-category construction with its degree-zero cohomology algebra, and would extend the algebraic result from quivers containing cycles to every connected non-Dynkin quiver. It is presented as an additional conjecture alongside the 2-Calabi–Yau conjecture.