Conjecture on stability from the canonical Poisson cohomology map

Let $(M,\pi)$ be a Poisson manifold and let $S$ be a compact symplectic leaf. Consider the canonical map

$$\Phi:H^{2}_{\pi,S}(M)\longrightarrow H^{2}_{\pi}(M,S).$$

Stability conjecture. If $\Phi$ vanishes, then any Poisson structure close enough to $\pi$ admits at least one nearby symplectic leaf diffeomorphic to $S$.

This conjecture proposes a relation between stability as a manifold and strong stability as a symplectic manifold. The preceding stability and strong stability theorems establish sufficient conditions using the vanishing of the target and source cohomology groups, respectively; to our knowledge, this conjecture is still open.