Rigidity conjecture for Hopf actions on Weyl Poisson algebras

Let $H$ be a finite-dimensional Hopf algebra acting on a Weyl Poisson algebra. Rigidity conjecture. The $H$-action must factor through a group algebra. This conjecture asks whether the rigidity theorem known for Weyl Poisson algebras under actions preserving the natural filtration remains valid without that filtration-preservation hypothesis; the analogous statement is known on the associative Weyl algebra side.