Bondal's conjecture on degeneracy loci of Fano Poisson manifolds

Let $(X,\pi)$ be a Fano Poisson complex manifold of rank $r$. Its degeneracy loci are

$$D_{2k}(\pi)=\{x\in X\mid \pi^{k+1}(x)=0\}.$$

Bondal's conjecture. For every $k\in\{0,1,\dots,r-1\}$, $D_{2k}(\pi)$ has an irreducible component of dimension at least $2k+1$.

The conjecture concerns the dimensions of the Poisson degeneracy loci and their role in stratifying a Fano Poisson manifold. This case was settled in the cited work of Polischuk.