2 problems

Poisson-pencil conjecture. There exists a pencil of compatible Poisson brackets on ${\mathcal M}_{n,d,q}^{\times}/\!/\operatorname{GL}_n(\mathbb C)$ containing the Arutyunov–Olivuc…

Invariant-integrals conjecture. The pair $(\mathfrak I,\mathfrak H_{\operatorname{tr}})$ yields a degenerate integrable system on $\Phi^{-1}(e^{\mathsf i\gamma})/{\rm U}(n)$.