Relative Poisson cohomology criterion for stable symplectic leaves

Let $M$ be a manifold with a Poisson structure $pi$, and let $S\subset M$ be a symplectic leaf. The relative Poisson cohomology of $S$ is the cohomology of the complex $({\mathfrak{X}}^k(M,S),\delta)$, where

$${\mathfrak{X}}^k(M,S)=\Gamma(\wedge^k T^*_S M),$$

and $\delta\theta=[\theta,pi]$; it is denoted by $H^\bullet_{\pi}(M,S)$. The leaf $S$ is stable if every tubular neighborhood of $S$ contains, for every sufficiently close Poisson structure, a leaf diffeomorphic to $S$.

Relative Poisson cohomology conjecture. Let $pi$ be a Poisson structure with a compact symplectic leaf $S\subset M$. If

$$H^2_{\pi}(M,S)=0,$$

then $S$ is a stable symplectic leaf.

For a zero-dimensional leaf this reduces to the fixed-point stability criterion in terms of the second cohomology of the isotropy Lie algebra. The conjecture is motivated by the ellipticity and finite-dimensionality of the relative Poisson cohomology for compact leaves; the source gives further evidence but does not state a resolution.