Euler continuant quasi-bisymplecticity conjecture for the localized doubled-quiver algebra

Let ${\mathcal{B}}(\Gamma_n)$ be the localization of the path algebra of the doubled quiver associated with $\Gamma_n$, let $\Phi$ be its multiplicative moment map, and let $\omega_n$ be the associated noncommutative $2$-form. The double bracket $\{\mkern-6mu\{-,-\}\mkern-6mu\}$ is the bracket defined by the Euler-continuant construction.

Quasi-bisymplecticity conjecture. The triple $({\mathcal{B}}(\Gamma_n),\Phi,\omega_n)$ is a quasi-bisymplectic algebra. Furthermore, the double quasi-Poisson bracket $\{\mkern-6mu\{-,-\}\mkern-6mu\}$ is non-degenerate and compatible with $\omega_n$.

Non-degeneracy of the double quasi-Poisson bracket was proved by Van den Bergh, while the remaining assertions were subsequently established by Bozec, Calaque and Scherotzke using relative Calabi–Yau structures. Thus the conjecture is solved.