Bondal's degeneracy-locus conjecture for Poisson structures on Fano varieties

Let $X$ be a connected Fano variety of dimension $n$ and let $\pi$ be a Poisson structure on it. For an integer $k\geq 0$ satisfying $2k<n$, define the degeneracy locus $X_{\leq 2k}:=\{x\in X\mid \operatorname{rk}_x\pi\leq 2k\}$. Bondal's conjecture. If $X_{\leq 2k}$ is not empty, then it contains a component of dimension at least $2k+1$. This conjecture predicts a dimension lower bound for nonempty degeneracy loci of Poisson structures on Fano varieties. The paper presents a toric analogue and proves related results for equivariant bi-vector fields, while the conjecture itself is attributed to Bondal.